| L(s) = 1 | + 2-s + 8·3-s + 3·4-s + 8·6-s + 7·8-s + 20·9-s − 32·11-s + 24·12-s − 9·16-s + 80·17-s + 20·18-s − 56·19-s − 32·22-s + 56·24-s + 92·25-s − 24·27-s − 5·32-s − 256·33-s + 80·34-s + 60·36-s − 56·38-s − 128·41-s − 96·44-s − 72·48-s + 92·50-s + 640·51-s − 24·54-s + ⋯ |
| L(s) = 1 | + 1/2·2-s + 8/3·3-s + 3/4·4-s + 4/3·6-s + 7/8·8-s + 20/9·9-s − 2.90·11-s + 2·12-s − 0.562·16-s + 4.70·17-s + 10/9·18-s − 2.94·19-s − 1.45·22-s + 7/3·24-s + 3.67·25-s − 8/9·27-s − 0.156·32-s − 7.75·33-s + 2.35·34-s + 5/3·36-s − 1.47·38-s − 3.12·41-s − 2.18·44-s − 3/2·48-s + 1.83·50-s + 12.5·51-s − 4/9·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(41.09287862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(41.09287862\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T - p T^{2} - p T^{3} + 3 p^{3} T^{4} - p^{3} T^{5} - p^{5} T^{6} - p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | \( 1 \) |
| good | 3 | \( ( 1 - 4 T + 14 T^{2} - 28 T^{3} + 98 T^{4} - 28 p^{2} T^{5} + 14 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 5 | \( 1 - 92 T^{2} + 5464 T^{4} - 211956 T^{6} + 6231214 T^{8} - 211956 p^{4} T^{10} + 5464 p^{8} T^{12} - 92 p^{12} T^{14} + p^{16} T^{16} \) |
| 11 | \( ( 1 + 16 T + 328 T^{2} + 4944 T^{3} + 49230 T^{4} + 4944 p^{2} T^{5} + 328 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 13 | \( 1 - 444 T^{2} + 142936 T^{4} - 35361044 T^{6} + 6575436334 T^{8} - 35361044 p^{4} T^{10} + 142936 p^{8} T^{12} - 444 p^{12} T^{14} + p^{16} T^{16} \) |
| 17 | \( ( 1 - 40 T + 1308 T^{2} - 31512 T^{3} + 588230 T^{4} - 31512 p^{2} T^{5} + 1308 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 + 28 T + 90 p T^{2} + 31332 T^{3} + 975266 T^{4} + 31332 p^{2} T^{5} + 90 p^{5} T^{6} + 28 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( 1 - 1744 T^{2} + 1272156 T^{4} - 412076080 T^{6} + 99307893702 T^{8} - 412076080 p^{4} T^{10} + 1272156 p^{8} T^{12} - 1744 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( 1 - 3384 T^{2} + 6555580 T^{4} - 8754768776 T^{6} + 8490907402822 T^{8} - 8754768776 p^{4} T^{10} + 6555580 p^{8} T^{12} - 3384 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( 1 - 3944 T^{2} + 8438620 T^{4} - 12447428312 T^{6} + 13694235978694 T^{8} - 12447428312 p^{4} T^{10} + 8438620 p^{8} T^{12} - 3944 p^{12} T^{14} + p^{16} T^{16} \) |
| 37 | \( 1 - 3512 T^{2} + 9188668 T^{4} - 18622781448 T^{6} + 27544347275206 T^{8} - 18622781448 p^{4} T^{10} + 9188668 p^{8} T^{12} - 3512 p^{12} T^{14} + p^{16} T^{16} \) |
| 41 | \( ( 1 + 64 T + 4956 T^{2} + 221760 T^{3} + 11848326 T^{4} + 221760 p^{2} T^{5} + 4956 p^{4} T^{6} + 64 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 + 4680 T^{2} - 58016 T^{3} + 10251086 T^{4} - 58016 p^{2} T^{5} + 4680 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 - 8392 T^{2} + 39566748 T^{4} - 127207295352 T^{6} + 316693927920198 T^{8} - 127207295352 p^{4} T^{10} + 39566748 p^{8} T^{12} - 8392 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( 1 - 18920 T^{2} + 162796828 T^{4} - 840091728600 T^{6} + 2864724835962118 T^{8} - 840091728600 p^{4} T^{10} + 162796828 p^{8} T^{12} - 18920 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( ( 1 + 52 T + 2254 T^{2} - 207508 T^{3} - 19795230 T^{4} - 207508 p^{2} T^{5} + 2254 p^{4} T^{6} + 52 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 61 | \( 1 - 16316 T^{2} + 140172120 T^{4} - 816942037524 T^{6} + 3499102878259502 T^{8} - 816942037524 p^{4} T^{10} + 140172120 p^{8} T^{12} - 16316 p^{12} T^{14} + p^{16} T^{16} \) |
| 67 | \( ( 1 - 152 T + 22224 T^{2} - 2037320 T^{3} + 158433022 T^{4} - 2037320 p^{2} T^{5} + 22224 p^{4} T^{6} - 152 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( 1 - 9864 T^{2} + 51888284 T^{4} - 294531431096 T^{6} + 1789421441990854 T^{8} - 294531431096 p^{4} T^{10} + 51888284 p^{8} T^{12} - 9864 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 - 56 T + 18460 T^{2} - 736008 T^{3} + 138223494 T^{4} - 736008 p^{2} T^{5} + 18460 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( 1 - 24968 T^{2} + 330869788 T^{4} - 3106127956152 T^{6} + 22189846569597766 T^{8} - 3106127956152 p^{4} T^{10} + 330869788 p^{8} T^{12} - 24968 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( ( 1 + 36 T + 16478 T^{2} + 177884 T^{3} + 135298114 T^{4} + 177884 p^{2} T^{5} + 16478 p^{4} T^{6} + 36 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 - 256 T + 48252 T^{2} - 6269952 T^{3} + 638304966 T^{4} - 6269952 p^{2} T^{5} + 48252 p^{4} T^{6} - 256 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 + 32 T + 19484 T^{2} + 1437536 T^{3} + 199130566 T^{4} + 1437536 p^{2} T^{5} + 19484 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.81819519697510930582548914022, −4.56010558050431289420835658383, −4.48923716203443472841476439037, −4.41316357802806415906117538770, −4.13649466747800631919062279387, −3.89768946721223058537500190063, −3.62601473433150878941801920387, −3.55200481050898116217077972269, −3.33307115006048643328231826525, −3.31237799257562587206282800389, −3.23421337797619913467664676033, −3.04422085338346678448965618689, −2.99138919944400368940478642214, −2.82653008503684393534480123922, −2.61083633661196776965423302115, −2.27483038840077298062432387204, −2.24963277963283552312940691819, −2.13364816315999880769503909251, −1.82554210452422785369315051359, −1.76757770966188732746711323815, −1.69779087256037813366955644484, −0.940363236563343918345989603544, −0.825190252622306324028297897711, −0.54293764238753195314848613684, −0.45820371883199201063167267547,
0.45820371883199201063167267547, 0.54293764238753195314848613684, 0.825190252622306324028297897711, 0.940363236563343918345989603544, 1.69779087256037813366955644484, 1.76757770966188732746711323815, 1.82554210452422785369315051359, 2.13364816315999880769503909251, 2.24963277963283552312940691819, 2.27483038840077298062432387204, 2.61083633661196776965423302115, 2.82653008503684393534480123922, 2.99138919944400368940478642214, 3.04422085338346678448965618689, 3.23421337797619913467664676033, 3.31237799257562587206282800389, 3.33307115006048643328231826525, 3.55200481050898116217077972269, 3.62601473433150878941801920387, 3.89768946721223058537500190063, 4.13649466747800631919062279387, 4.41316357802806415906117538770, 4.48923716203443472841476439037, 4.56010558050431289420835658383, 4.81819519697510930582548914022
Plot not available for L-functions of degree greater than 10.
