| L(s) = 1 | + 5·2-s + 7·3-s + 17·4-s + 35·6-s + 13·7-s + 45·8-s + 22·9-s − 26·11-s + 119·12-s − 13·13-s + 65·14-s + 89·16-s − 77·17-s + 110·18-s − 126·19-s + 91·21-s − 130·22-s + 96·23-s + 315·24-s − 65·26-s − 35·27-s + 221·28-s − 82·29-s + 196·31-s + 85·32-s − 182·33-s − 385·34-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 1.34·3-s + 17/8·4-s + 2.38·6-s + 0.701·7-s + 1.98·8-s + 0.814·9-s − 0.712·11-s + 2.86·12-s − 0.277·13-s + 1.24·14-s + 1.39·16-s − 1.09·17-s + 1.44·18-s − 1.52·19-s + 0.945·21-s − 1.25·22-s + 0.870·23-s + 2.67·24-s − 0.490·26-s − 0.249·27-s + 1.49·28-s − 0.525·29-s + 1.13·31-s + 0.469·32-s − 0.960·33-s − 1.94·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.843411595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.843411595\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + p T \) |
| good | 2 | \( 1 - 5 T + p^{3} T^{2} \) |
| 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 7 | \( 1 - 13 T + p^{3} T^{2} \) |
| 11 | \( 1 + 26 T + p^{3} T^{2} \) |
| 17 | \( 1 + 77 T + p^{3} T^{2} \) |
| 19 | \( 1 + 126 T + p^{3} T^{2} \) |
| 23 | \( 1 - 96 T + p^{3} T^{2} \) |
| 29 | \( 1 + 82 T + p^{3} T^{2} \) |
| 31 | \( 1 - 196 T + p^{3} T^{2} \) |
| 37 | \( 1 - 131 T + p^{3} T^{2} \) |
| 41 | \( 1 - 336 T + p^{3} T^{2} \) |
| 43 | \( 1 - 201 T + p^{3} T^{2} \) |
| 47 | \( 1 - 105 T + p^{3} T^{2} \) |
| 53 | \( 1 - 432 T + p^{3} T^{2} \) |
| 59 | \( 1 + 294 T + p^{3} T^{2} \) |
| 61 | \( 1 + 56 T + p^{3} T^{2} \) |
| 67 | \( 1 + 478 T + p^{3} T^{2} \) |
| 71 | \( 1 - 9 T + p^{3} T^{2} \) |
| 73 | \( 1 + 98 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1304 T + p^{3} T^{2} \) |
| 83 | \( 1 - 308 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1190 T + p^{3} T^{2} \) |
| 97 | \( 1 + 70 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.32632539663230325888854575716, −10.64103410671991889828601947453, −9.121538292189049803097595760717, −8.157704225583179005068261181591, −7.23797986806393998417624478761, −6.08202034007153900949799328261, −4.78983132239334595044417180735, −4.09337497006856782755211380394, −2.76741512480276335452011428509, −2.14493965042283773697129531838,
2.14493965042283773697129531838, 2.76741512480276335452011428509, 4.09337497006856782755211380394, 4.78983132239334595044417180735, 6.08202034007153900949799328261, 7.23797986806393998417624478761, 8.157704225583179005068261181591, 9.121538292189049803097595760717, 10.64103410671991889828601947453, 11.32632539663230325888854575716
