| L(s) = 1 | − 8·4-s + 36·16-s − 16·25-s − 32·37-s + 16·43-s − 120·64-s + 16·67-s + 128·100-s − 64·109-s + 96·121-s + 127-s + 131-s + 137-s + 139-s + 256·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 152·169-s − 128·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 4·4-s + 9·16-s − 3.19·25-s − 5.26·37-s + 2.43·43-s − 15·64-s + 1.95·67-s + 64/5·100-s − 6.13·109-s + 8.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 21.0·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 11.6·169-s − 9.75·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{48} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{48} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.108677203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.108677203\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{8} \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 + 8 T^{2} + 24 T^{4} + 8 p T^{6} + 146 T^{8} + 8 p^{3} T^{10} + 24 p^{4} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 11 | \( ( 1 - 48 T^{2} + 1120 T^{4} - 17712 T^{6} + 217314 T^{8} - 17712 p^{2} T^{10} + 1120 p^{4} T^{12} - 48 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 13 | \( ( 1 - 76 T^{2} + 210 p T^{4} - 61328 T^{6} + 948659 T^{8} - 61328 p^{2} T^{10} + 210 p^{5} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 17 | \( ( 1 + 52 T^{2} + 1414 T^{4} + 26440 T^{6} + 429547 T^{8} + 26440 p^{2} T^{10} + 1414 p^{4} T^{12} + 52 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 19 | \( ( 1 - 72 T^{2} + 2232 T^{4} - 42216 T^{6} + 719954 T^{8} - 42216 p^{2} T^{10} + 2232 p^{4} T^{12} - 72 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 23 | \( ( 1 - 68 T^{2} + 3226 T^{4} - 98288 T^{6} + 2591299 T^{8} - 98288 p^{2} T^{10} + 3226 p^{4} T^{12} - 68 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 29 | \( ( 1 - 172 T^{2} + 14214 T^{4} - 731096 T^{6} + 25478795 T^{8} - 731096 p^{2} T^{10} + 14214 p^{4} T^{12} - 172 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 31 | \( ( 1 - 172 T^{2} + 14214 T^{4} - 743480 T^{6} + 27186923 T^{8} - 743480 p^{2} T^{10} + 14214 p^{4} T^{12} - 172 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 37 | \( ( 1 + 8 T + 52 T^{2} + 440 T^{3} + 4174 T^{4} + 440 p T^{5} + 52 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 + 176 T^{2} + 18132 T^{4} + 1209232 T^{6} + 58623014 T^{8} + 1209232 p^{2} T^{10} + 18132 p^{4} T^{12} + 176 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 43 | \( ( 1 - 4 T + 18 T^{2} - 32 T^{3} + 2417 T^{4} - 32 p T^{5} + 18 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 47 | \( ( 1 + 224 T^{2} + 25344 T^{4} + 1919584 T^{6} + 2239006 p T^{8} + 1919584 p^{2} T^{10} + 25344 p^{4} T^{12} + 224 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 53 | \( ( 1 - 228 T^{2} + 26422 T^{4} - 2159208 T^{6} + 132545883 T^{8} - 2159208 p^{2} T^{10} + 26422 p^{4} T^{12} - 228 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 59 | \( ( 1 + 204 T^{2} + 25242 T^{4} + 2082288 T^{6} + 138722051 T^{8} + 2082288 p^{2} T^{10} + 25242 p^{4} T^{12} + 204 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 61 | \( ( 1 - 184 T^{2} + 17340 T^{4} - 1108040 T^{6} + 62404262 T^{8} - 1108040 p^{2} T^{10} + 17340 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 67 | \( ( 1 - 4 T + 118 T^{2} - 496 T^{3} + 11881 T^{4} - 496 p T^{5} + 118 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \) |
| 71 | \( ( 1 - 508 T^{2} + 116778 T^{4} - 15836144 T^{6} + 1384193075 T^{8} - 15836144 p^{2} T^{10} + 116778 p^{4} T^{12} - 508 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 73 | \( ( 1 - 96 T^{2} + 4608 T^{4} - 471648 T^{6} + 60702530 T^{8} - 471648 p^{2} T^{10} + 4608 p^{4} T^{12} - 96 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 79 | \( ( 1 + 156 T^{2} + 120 T^{3} + 12548 T^{4} + 120 p T^{5} + 156 p^{2} T^{6} + p^{4} T^{8} )^{4} \) |
| 83 | \( ( 1 + 336 T^{2} + 49092 T^{4} + 4266096 T^{6} + 321975206 T^{8} + 4266096 p^{2} T^{10} + 49092 p^{4} T^{12} + 336 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 89 | \( ( 1 + 564 T^{2} + 1686 p T^{4} + 24367944 T^{6} + 2630246603 T^{8} + 24367944 p^{2} T^{10} + 1686 p^{5} T^{12} + 564 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
| 97 | \( ( 1 - 24 T^{2} + 29388 T^{4} - 211176 T^{6} + 370008998 T^{8} - 211176 p^{2} T^{10} + 29388 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.15418553149921814339146378194, −2.06685119517252813127655663897, −2.06152763726508655517551609463, −2.01220939405914222749702859628, −1.85236720427055038110076692805, −1.82521182315970535529942616180, −1.66843057924600281786028555903, −1.57975763895580529726796614126, −1.52906794429607114897296795673, −1.50385856239971373148209771966, −1.47125368798657787563951326978, −1.33394808517205169816291375013, −1.33338622094720583021153941667, −1.21617275486729990479977364230, −0.973369839164618990396853668811, −0.927368611801620316097927932768, −0.827683382890789604134482773524, −0.75314497027490316704780009808, −0.74712356187654814335155872681, −0.71373519968225971097398030494, −0.37247281663164837881946481398, −0.32903241109688261431490773067, −0.27309741594948712048088374875, −0.20646726011263744404295227837, −0.14166111798285187218648087728,
0.14166111798285187218648087728, 0.20646726011263744404295227837, 0.27309741594948712048088374875, 0.32903241109688261431490773067, 0.37247281663164837881946481398, 0.71373519968225971097398030494, 0.74712356187654814335155872681, 0.75314497027490316704780009808, 0.827683382890789604134482773524, 0.927368611801620316097927932768, 0.973369839164618990396853668811, 1.21617275486729990479977364230, 1.33338622094720583021153941667, 1.33394808517205169816291375013, 1.47125368798657787563951326978, 1.50385856239971373148209771966, 1.52906794429607114897296795673, 1.57975763895580529726796614126, 1.66843057924600281786028555903, 1.82521182315970535529942616180, 1.85236720427055038110076692805, 2.01220939405914222749702859628, 2.06152763726508655517551609463, 2.06685119517252813127655663897, 2.15418553149921814339146378194
Plot not available for L-functions of degree greater than 10.
