L(s) = 1 | + 1.72e3·2-s + 5.90e4·3-s + 8.88e5·4-s + 1.02e8·6-s − 3.15e8·7-s − 2.08e9·8-s + 3.48e9·9-s − 4.66e10·11-s + 5.24e10·12-s + 7.73e11·13-s − 5.45e11·14-s − 5.47e12·16-s + 8.80e12·17-s + 6.02e12·18-s − 2.41e13·19-s − 1.86e13·21-s − 8.05e13·22-s + 1.52e13·23-s − 1.23e14·24-s + 1.33e15·26-s + 2.05e14·27-s − 2.80e14·28-s − 2.37e15·29-s + 5.25e15·31-s − 5.07e15·32-s − 2.75e15·33-s + 1.52e16·34-s + ⋯ |
L(s) = 1 | + 1.19·2-s + 0.577·3-s + 0.423·4-s + 0.688·6-s − 0.422·7-s − 0.687·8-s + 0.333·9-s − 0.541·11-s + 0.244·12-s + 1.55·13-s − 0.504·14-s − 1.24·16-s + 1.05·17-s + 0.397·18-s − 0.903·19-s − 0.243·21-s − 0.646·22-s + 0.0765·23-s − 0.396·24-s + 1.85·26-s + 0.192·27-s − 0.179·28-s − 1.04·29-s + 1.15·31-s − 0.797·32-s − 0.312·33-s + 1.26·34-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 | \( 1 - 1.72e3T + 2.09e6T^{2} \) |
| 7 | \( 1 + 3.15e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 4.66e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 7.73e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 8.80e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 2.41e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 1.52e13T + 3.94e28T^{2} \) |
| 29 | \( 1 + 2.37e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 5.25e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 2.94e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 1.01e17T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.67e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 3.47e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 7.88e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 3.82e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 5.59e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.79e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 8.97e18T + 7.52e38T^{2} \) |
| 73 | \( 1 - 2.61e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.46e20T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.51e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 4.37e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 7.85e20T + 5.27e41T^{2} \) |
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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
−10.14969949818328389089924185493, −8.955618503401811758239289842027, −7.999385900899351219940586490662, −6.50811755733705297205852412225, −5.70064865670457900265843323527, −4.47093487617892711706546399988, −3.52507060448807268420257068939, −2.84797568146080495169995729879, −1.46052193754173532013913941111, 0,
1.46052193754173532013913941111, 2.84797568146080495169995729879, 3.52507060448807268420257068939, 4.47093487617892711706546399988, 5.70064865670457900265843323527, 6.50811755733705297205852412225, 7.999385900899351219940586490662, 8.955618503401811758239289842027, 10.14969949818328389089924185493