| L(s) = 1 | + 2.38e3·2-s + 5.90e4·3-s + 3.60e6·4-s + 1.40e8·6-s + 8.76e8·7-s + 3.59e9·8-s + 3.48e9·9-s − 1.10e11·11-s + 2.12e11·12-s − 6.22e11·13-s + 2.09e12·14-s + 1.02e12·16-s − 6.51e12·17-s + 8.32e12·18-s − 4.44e13·19-s + 5.17e13·21-s − 2.63e14·22-s + 2.65e14·23-s + 2.12e14·24-s − 1.48e15·26-s + 2.05e14·27-s + 3.15e15·28-s − 1.28e15·29-s − 5.90e15·31-s − 5.08e15·32-s − 6.52e15·33-s − 1.55e16·34-s + ⋯ |
| L(s) = 1 | + 1.64·2-s + 0.577·3-s + 1.71·4-s + 0.951·6-s + 1.17·7-s + 1.18·8-s + 0.333·9-s − 1.28·11-s + 0.992·12-s − 1.25·13-s + 1.93·14-s + 0.234·16-s − 0.783·17-s + 0.549·18-s − 1.66·19-s + 0.676·21-s − 2.11·22-s + 1.33·23-s + 0.683·24-s − 2.06·26-s + 0.192·27-s + 2.01·28-s − 0.565·29-s − 1.29·31-s − 0.798·32-s − 0.741·33-s − 1.29·34-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
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 - 2.38e3T + 2.09e6T^{2} \) |
| 7 | \( 1 - 8.76e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 1.10e11T + 7.40e21T^{2} \) |
| 13 | \( 1 + 6.22e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 6.51e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 4.44e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 2.65e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 1.28e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 5.90e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 3.24e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 4.85e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 5.89e16T + 2.00e34T^{2} \) |
| 47 | \( 1 + 3.80e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 7.98e17T + 1.62e36T^{2} \) |
| 59 | \( 1 - 3.84e17T + 1.54e37T^{2} \) |
| 61 | \( 1 - 3.61e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 9.40e18T + 2.22e38T^{2} \) |
| 71 | \( 1 - 8.73e18T + 7.52e38T^{2} \) |
| 73 | \( 1 - 4.81e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 8.49e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 1.25e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 1.17e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 1.21e21T + 5.27e41T^{2} \) |
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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
−10.55762013643741349361596479105, −8.858005463050609216860368710028, −7.69951695156995148894112398870, −6.75363998724664152983199067178, −5.15833338477426904556008968999, −4.84755122024737702505413230512, −3.67097647143810094880010152434, −2.40906601755788218830746599092, −2.00102942477157535951270343015, 0,
2.00102942477157535951270343015, 2.40906601755788218830746599092, 3.67097647143810094880010152434, 4.84755122024737702505413230512, 5.15833338477426904556008968999, 6.75363998724664152983199067178, 7.69951695156995148894112398870, 8.858005463050609216860368710028, 10.55762013643741349361596479105
