| L(s) = 1 | + 5·2-s + 17·4-s + 14·5-s − 32·7-s + 45·8-s + 70·10-s + 11·11-s − 38·13-s − 160·14-s + 89·16-s + 2·17-s + 72·19-s + 238·20-s + 55·22-s − 68·23-s + 71·25-s − 190·26-s − 544·28-s + 54·29-s − 152·31-s + 85·32-s + 10·34-s − 448·35-s + 174·37-s + 360·38-s + 630·40-s − 94·41-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 17/8·4-s + 1.25·5-s − 1.72·7-s + 1.98·8-s + 2.21·10-s + 0.301·11-s − 0.810·13-s − 3.05·14-s + 1.39·16-s + 0.0285·17-s + 0.869·19-s + 2.66·20-s + 0.533·22-s − 0.616·23-s + 0.567·25-s − 1.43·26-s − 3.67·28-s + 0.345·29-s − 0.880·31-s + 0.469·32-s + 0.0504·34-s − 2.16·35-s + 0.773·37-s + 1.53·38-s + 2.49·40-s − 0.358·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.118740257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.118740257\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - p T \) |
| good | 2 | \( 1 - 5 T + p^{3} T^{2} \) |
| 5 | \( 1 - 14 T + p^{3} T^{2} \) |
| 7 | \( 1 + 32 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 2 T + p^{3} T^{2} \) |
| 19 | \( 1 - 72 T + p^{3} T^{2} \) |
| 23 | \( 1 + 68 T + p^{3} T^{2} \) |
| 29 | \( 1 - 54 T + p^{3} T^{2} \) |
| 31 | \( 1 + 152 T + p^{3} T^{2} \) |
| 37 | \( 1 - 174 T + p^{3} T^{2} \) |
| 41 | \( 1 + 94 T + p^{3} T^{2} \) |
| 43 | \( 1 + 528 T + p^{3} T^{2} \) |
| 47 | \( 1 - 340 T + p^{3} T^{2} \) |
| 53 | \( 1 - 438 T + p^{3} T^{2} \) |
| 59 | \( 1 + 20 T + p^{3} T^{2} \) |
| 61 | \( 1 - 570 T + p^{3} T^{2} \) |
| 67 | \( 1 + 460 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1092 T + p^{3} T^{2} \) |
| 73 | \( 1 - 562 T + p^{3} T^{2} \) |
| 79 | \( 1 + 16 T + p^{3} T^{2} \) |
| 83 | \( 1 + 372 T + p^{3} T^{2} \) |
| 89 | \( 1 - 966 T + p^{3} T^{2} \) |
| 97 | \( 1 + 526 T + p^{3} T^{2} \) |
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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
−13.43434910137316612141022374741, −12.71125030925909594058041984741, −11.83235099854909654277750922131, −10.21423808108031227871944916599, −9.454680857221027075949373686511, −7.02366367103417308769025902453, −6.18528858126133271259285952533, −5.29499404212907390026734482434, −3.60958730236608261490424709415, −2.40684343637818605205952601774,
2.40684343637818605205952601774, 3.60958730236608261490424709415, 5.29499404212907390026734482434, 6.18528858126133271259285952533, 7.02366367103417308769025902453, 9.454680857221027075949373686511, 10.21423808108031227871944916599, 11.83235099854909654277750922131, 12.71125030925909594058041984741, 13.43434910137316612141022374741
