| L(s) = 1 | − 3·2-s − 7·4-s + 25·5-s − 78·7-s + 69·8-s + 81·9-s − 75·10-s + 121·11-s + 162·13-s + 234·14-s − 95·16-s + 402·17-s − 243·18-s − 175·20-s − 363·22-s + 625·25-s − 486·26-s + 546·28-s − 1.59e3·31-s − 819·32-s − 1.20e3·34-s − 1.95e3·35-s − 567·36-s + 1.72e3·40-s + 3.52e3·43-s − 847·44-s + 2.02e3·45-s + ⋯ |
| L(s) = 1 | − 3/4·2-s − 0.437·4-s + 5-s − 1.59·7-s + 1.07·8-s + 9-s − 3/4·10-s + 11-s + 0.958·13-s + 1.19·14-s − 0.371·16-s + 1.39·17-s − 3/4·18-s − 0.437·20-s − 3/4·22-s + 25-s − 0.718·26-s + 0.696·28-s − 1.66·31-s − 0.799·32-s − 1.04·34-s − 1.59·35-s − 0.437·36-s + 1.07·40-s + 1.90·43-s − 0.437·44-s + 45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.090093156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.090093156\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - p^{2} T \) |
| 11 | \( 1 - p^{2} T \) |
| good | 2 | \( 1 + 3 T + p^{4} T^{2} \) |
| 3 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 7 | \( 1 + 78 T + p^{4} T^{2} \) |
| 13 | \( 1 - 162 T + p^{4} T^{2} \) |
| 17 | \( 1 - 402 T + p^{4} T^{2} \) |
| 19 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 31 | \( 1 + 1598 T + p^{4} T^{2} \) |
| 37 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( 1 - 3522 T + p^{4} T^{2} \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 59 | \( 1 - 3442 T + p^{4} T^{2} \) |
| 61 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 67 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 71 | \( 1 + 3998 T + p^{4} T^{2} \) |
| 73 | \( 1 + 10638 T + p^{4} T^{2} \) |
| 79 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 83 | \( 1 - 13602 T + p^{4} T^{2} \) |
| 89 | \( 1 + 15838 T + p^{4} T^{2} \) |
| 97 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
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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
−14.37037296408660213649676044466, −13.30806682235148526042100439220, −12.60188675466443481024029637777, −10.50010570175545202432239091031, −9.656419559294173182443187840980, −9.041619573360570007835362288547, −7.16804240128046363763499258564, −5.88762702237267445877934833591, −3.77131252371471343448595855825, −1.20089647082121139804533191441,
1.20089647082121139804533191441, 3.77131252371471343448595855825, 5.88762702237267445877934833591, 7.16804240128046363763499258564, 9.041619573360570007835362288547, 9.656419559294173182443187840980, 10.50010570175545202432239091031, 12.60188675466443481024029637777, 13.30806682235148526042100439220, 14.37037296408660213649676044466
