| L(s) = 1 | + 9·3-s + 25·5-s + 81·9-s + 238·11-s + 142·13-s + 225·15-s − 98·17-s − 862·23-s + 625·25-s + 729·27-s − 238·29-s − 1.44e3·31-s + 2.14e3·33-s + 1.58e3·37-s + 1.27e3·39-s + 1.77e3·43-s + 2.02e3·45-s − 3.26e3·47-s + 2.40e3·49-s − 882·51-s + 5.95e3·55-s − 2.64e3·59-s + 3.55e3·65-s − 8.30e3·67-s − 7.75e3·69-s + 5.62e3·75-s + 1.10e4·79-s + ⋯ |
| L(s) = 1 | + 3-s + 5-s + 9-s + 1.96·11-s + 0.840·13-s + 15-s − 0.339·17-s − 1.62·23-s + 25-s + 27-s − 0.282·29-s − 1.50·31-s + 1.96·33-s + 1.15·37-s + 0.840·39-s + 0.961·43-s + 45-s − 1.47·47-s + 49-s − 0.339·51-s + 1.96·55-s − 0.758·59-s + 0.840·65-s − 1.84·67-s − 1.62·69-s + 75-s + 1.76·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(4.372188602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.372188602\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| 5 | \( 1 - p^{2} T \) |
| good | 7 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 11 | \( 1 - 238 T + p^{4} T^{2} \) |
| 13 | \( 1 - 142 T + p^{4} T^{2} \) |
| 17 | \( 1 + 98 T + p^{4} T^{2} \) |
| 19 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 23 | \( 1 + 862 T + p^{4} T^{2} \) |
| 29 | \( 1 + 238 T + p^{4} T^{2} \) |
| 31 | \( 1 + 1442 T + p^{4} T^{2} \) |
| 37 | \( 1 - 1582 T + p^{4} T^{2} \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( 1 - 1778 T + p^{4} T^{2} \) |
| 47 | \( 1 + 3262 T + p^{4} T^{2} \) |
| 53 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 59 | \( 1 + 2642 T + p^{4} T^{2} \) |
| 61 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 67 | \( 1 + 8302 T + p^{4} T^{2} \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 79 | \( 1 - 11038 T + p^{4} T^{2} \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 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
−10.13810767448366794697188482196, −9.220047292877119396712604833169, −8.956856236154643431992737755778, −7.74438848046424552404400296787, −6.59847999906604929541710984241, −5.95067648145994342187808421444, −4.32366940995637561406119201394, −3.52842011400937727639494768814, −2.08520699033568613387297504575, −1.29247329609176787118035460868,
1.29247329609176787118035460868, 2.08520699033568613387297504575, 3.52842011400937727639494768814, 4.32366940995637561406119201394, 5.95067648145994342187808421444, 6.59847999906604929541710984241, 7.74438848046424552404400296787, 8.956856236154643431992737755778, 9.220047292877119396712604833169, 10.13810767448366794697188482196
