| L(s) = 1 | + 18·3-s − 8·4-s + 28·7-s + 123·9-s − 54·11-s − 144·12-s − 918·17-s + 30·19-s + 504·21-s + 486·23-s + 270·27-s − 224·28-s + 3.24e3·29-s − 546·31-s − 972·33-s − 984·36-s + 446·37-s − 2.34e3·43-s + 432·44-s − 702·47-s − 4.21e3·49-s − 1.65e4·51-s − 2.75e3·53-s + 540·57-s + 1.23e4·59-s + 7.68e3·61-s + 3.44e3·63-s + ⋯ |
| L(s) = 1 | + 2·3-s − 1/2·4-s + 4/7·7-s + 1.51·9-s − 0.446·11-s − 12-s − 3.17·17-s + 0.0831·19-s + 8/7·21-s + 0.918·23-s + 0.370·27-s − 2/7·28-s + 3.85·29-s − 0.568·31-s − 0.892·33-s − 0.759·36-s + 0.325·37-s − 1.26·43-s + 0.223·44-s − 0.317·47-s − 1.75·49-s − 6.35·51-s − 0.980·53-s + 0.166·57-s + 3.55·59-s + 2.06·61-s + 0.867·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(13.90624890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.90624890\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 + p^{3} T^{2} + p^{6} T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 2 p T + p^{4} T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 p^{2} T + 67 p T^{2} - 62 p^{3} T^{3} + 1204 p^{2} T^{4} - 62 p^{7} T^{5} + 67 p^{9} T^{6} - 2 p^{14} T^{7} + p^{16} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 54 T - 2437 p T^{2} + 23814 T^{3} + 626404692 T^{4} + 23814 p^{4} T^{5} - 2437 p^{9} T^{6} + 54 p^{12} T^{7} + p^{16} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 51652 T^{2} + 2274555846 T^{4} - 51652 p^{8} T^{6} + p^{16} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 54 p T + 493601 T^{2} + 11485422 p T^{3} + 61724271876 T^{4} + 11485422 p^{5} T^{5} + 493601 p^{8} T^{6} + 54 p^{13} T^{7} + p^{16} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 30 T + 66617 T^{2} - 1989510 T^{3} - 12546522252 T^{4} - 1989510 p^{4} T^{5} + 66617 p^{8} T^{6} - 30 p^{12} T^{7} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 486 T + 78265 T^{2} + 195250986 T^{3} - 119466109356 T^{4} + 195250986 p^{4} T^{5} + 78265 p^{8} T^{6} - 486 p^{12} T^{7} + p^{16} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 1620 T + 2066054 T^{2} - 1620 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 546 T + 1193657 T^{2} + 597479610 T^{3} + 436340752596 T^{4} + 597479610 p^{4} T^{5} + 1193657 p^{8} T^{6} + 546 p^{12} T^{7} + p^{16} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 446 T - 3015935 T^{2} + 237928066 T^{3} + 6449986888804 T^{4} + 237928066 p^{4} T^{5} - 3015935 p^{8} T^{6} - 446 p^{12} T^{7} + p^{16} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 9195268 T^{2} + 37043496050310 T^{4} - 9195268 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 1172 T + 3821766 T^{2} + 1172 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 702 T + 7568153 T^{2} + 5197527270 T^{3} + 31807801869972 T^{4} + 5197527270 p^{4} T^{5} + 7568153 p^{8} T^{6} + 702 p^{12} T^{7} + p^{16} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 2754 T - 5299967 T^{2} - 7976903166 T^{3} + 43904732373732 T^{4} - 7976903166 p^{4} T^{5} - 5299967 p^{8} T^{6} + 2754 p^{12} T^{7} + p^{16} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 12366 T + 87438953 T^{2} - 450942278166 T^{3} + 1800614696429652 T^{4} - 450942278166 p^{4} T^{5} + 87438953 p^{8} T^{6} - 12366 p^{12} T^{7} + p^{16} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 126 p T + 39234641 T^{2} - 2462431734 p T^{3} + 462871617507012 T^{4} - 2462431734 p^{5} T^{5} + 39234641 p^{8} T^{6} - 126 p^{13} T^{7} + p^{16} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 5062 T + 1333849 T^{2} + 81053994314 T^{3} - 332413001385740 T^{4} + 81053994314 p^{4} T^{5} + 1333849 p^{8} T^{6} - 5062 p^{12} T^{7} + p^{16} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 9396 T + 71231078 T^{2} - 9396 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 17274 T + 161686097 T^{2} - 1074829823970 T^{3} + 5889761488255716 T^{4} - 1074829823970 p^{4} T^{5} + 161686097 p^{8} T^{6} - 17274 p^{12} T^{7} + p^{16} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 794 T - 69869063 T^{2} + 5876126422 T^{3} + 3428515016079124 T^{4} + 5876126422 p^{4} T^{5} - 69869063 p^{8} T^{6} - 794 p^{12} T^{7} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 153397060 T^{2} + 10369989980918982 T^{4} - 153397060 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12474 T + 182683793 T^{2} + 1631810023074 T^{3} + 16430717819326692 T^{4} + 1631810023074 p^{4} T^{5} + 182683793 p^{8} T^{6} + 12474 p^{12} T^{7} + p^{16} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 281924740 T^{2} + 35525126061727494 T^{4} - 281924740 p^{8} T^{6} + p^{16} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.75654912680390392580046863288, −7.72051952011568126399211396622, −6.98472061411123576409426767294, −6.83763277512858739333988392375, −6.81346538295956125725552868785, −6.54753768477398721434441477944, −6.40401939617012242199519146722, −5.91025491214455960177142006862, −5.36946388895995145349277064751, −5.11372807219003564568781986815, −4.92958585513991872576536280617, −4.78147495669457505772075086497, −4.33074708031180550508761990079, −4.31743923875119783719428080164, −3.79167361036181296797326050207, −3.35709930483157078113187731927, −3.15692314496346121154492456982, −3.11147375819959749583559441478, −2.27471809025607741032863747033, −2.23502280431539355097113605312, −2.18047121442766231484957039078, −1.86583663188893060041048135667, −0.794910042265552653217987305491, −0.75438683548745166603941552326, −0.53247139257723333468682156458,
0.53247139257723333468682156458, 0.75438683548745166603941552326, 0.794910042265552653217987305491, 1.86583663188893060041048135667, 2.18047121442766231484957039078, 2.23502280431539355097113605312, 2.27471809025607741032863747033, 3.11147375819959749583559441478, 3.15692314496346121154492456982, 3.35709930483157078113187731927, 3.79167361036181296797326050207, 4.31743923875119783719428080164, 4.33074708031180550508761990079, 4.78147495669457505772075086497, 4.92958585513991872576536280617, 5.11372807219003564568781986815, 5.36946388895995145349277064751, 5.91025491214455960177142006862, 6.40401939617012242199519146722, 6.54753768477398721434441477944, 6.81346538295956125725552868785, 6.83763277512858739333988392375, 6.98472061411123576409426767294, 7.72051952011568126399211396622, 7.75654912680390392580046863288
