| L(s) = 1 | + 1.29e3·2-s + 5.90e4·3-s − 4.24e5·4-s + 7.63e7·6-s − 1.27e9·7-s − 3.26e9·8-s + 3.48e9·9-s − 1.21e11·11-s − 2.50e10·12-s − 7.37e11·13-s − 1.65e12·14-s − 3.32e12·16-s − 6.05e12·17-s + 4.50e12·18-s − 3.23e13·19-s − 7.55e13·21-s − 1.56e14·22-s + 3.98e13·23-s − 1.92e14·24-s − 9.53e14·26-s + 2.05e14·27-s + 5.43e14·28-s + 1.40e15·29-s − 2.62e15·31-s + 2.53e15·32-s − 7.15e15·33-s − 7.83e15·34-s + ⋯ |
| L(s) = 1 | + 0.893·2-s + 0.577·3-s − 0.202·4-s + 0.515·6-s − 1.71·7-s − 1.07·8-s + 0.333·9-s − 1.40·11-s − 0.116·12-s − 1.48·13-s − 1.52·14-s − 0.756·16-s − 0.728·17-s + 0.297·18-s − 1.20·19-s − 0.988·21-s − 1.25·22-s + 0.200·23-s − 0.619·24-s − 1.32·26-s + 0.192·27-s + 0.346·28-s + 0.622·29-s − 0.575·31-s + 0.398·32-s − 0.813·33-s − 0.650·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(0.1723404673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1723404673\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 5.90e4T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.29e3T + 2.09e6T^{2} \) |
| 7 | \( 1 + 1.27e9T + 5.58e17T^{2} \) |
| 11 | \( 1 + 1.21e11T + 7.40e21T^{2} \) |
| 13 | \( 1 + 7.37e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 6.05e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 3.23e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 3.98e13T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.40e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 2.62e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 3.44e15T + 8.55e32T^{2} \) |
| 41 | \( 1 - 7.99e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 2.07e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 1.95e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 2.33e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 3.93e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 8.99e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 9.21e18T + 2.22e38T^{2} \) |
| 71 | \( 1 + 4.75e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 1.81e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.11e20T + 7.08e39T^{2} \) |
| 83 | \( 1 - 3.39e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 3.38e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 1.12e20T + 5.27e41T^{2} \) |
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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
−10.41218431125928276810341548144, −9.602143859462294061653295592185, −8.637172717200705889194547998953, −7.23288116107843061892464492714, −6.20575981451759787092153793599, −5.03779278453668928634461865772, −4.06310420032187211214261249356, −2.89478536220428508178481633179, −2.46829729830096611179535503780, −0.13215812560812576347308888797,
0.13215812560812576347308888797, 2.46829729830096611179535503780, 2.89478536220428508178481633179, 4.06310420032187211214261249356, 5.03779278453668928634461865772, 6.20575981451759787092153793599, 7.23288116107843061892464492714, 8.637172717200705889194547998953, 9.602143859462294061653295592185, 10.41218431125928276810341548144
