| L(s) = 1 | − 2.54e3·2-s − 5.90e4·3-s + 4.35e6·4-s + 1.50e8·6-s − 4.55e8·7-s − 5.74e9·8-s + 3.48e9·9-s − 1.12e10·11-s − 2.57e11·12-s − 8.05e11·13-s + 1.15e12·14-s + 5.44e12·16-s + 3.31e11·17-s − 8.85e12·18-s + 1.86e13·19-s + 2.68e13·21-s + 2.85e13·22-s − 4.27e13·23-s + 3.39e14·24-s + 2.04e15·26-s − 2.05e14·27-s − 1.98e15·28-s + 3.70e15·29-s − 7.15e15·31-s − 1.80e15·32-s + 6.63e14·33-s − 8.41e14·34-s + ⋯ |
| L(s) = 1 | − 1.75·2-s − 0.577·3-s + 2.07·4-s + 1.01·6-s − 0.608·7-s − 1.89·8-s + 0.333·9-s − 0.130·11-s − 1.19·12-s − 1.61·13-s + 1.06·14-s + 1.23·16-s + 0.0398·17-s − 0.584·18-s + 0.697·19-s + 0.351·21-s + 0.229·22-s − 0.214·23-s + 1.09·24-s + 2.84·26-s − 0.192·27-s − 1.26·28-s + 1.63·29-s − 1.56·31-s − 0.283·32-s + 0.0753·33-s − 0.0699·34-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 | \( 1 + 2.54e3T + 2.09e6T^{2} \) |
| 7 | \( 1 + 4.55e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 1.12e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 8.05e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 3.31e11T + 6.90e25T^{2} \) |
| 19 | \( 1 - 1.86e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 4.27e13T + 3.94e28T^{2} \) |
| 29 | \( 1 - 3.70e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 7.15e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 2.94e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 4.25e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.81e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 3.00e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 2.29e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 6.29e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 7.85e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 9.53e18T + 2.22e38T^{2} \) |
| 71 | \( 1 - 1.28e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 7.89e18T + 1.34e39T^{2} \) |
| 79 | \( 1 + 7.72e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 2.36e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 1.56e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 1.41e21T + 5.27e41T^{2} \) |
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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
−9.859242716278973788015217426737, −9.369105439194281653412679256823, −7.957858299218419687105484165076, −7.17386994725067211127780975418, −6.23842269550892094546688358687, −4.86059228013340531983625673998, −3.02095114450699448904814605772, −1.96343175165254905925358599741, −0.76476867949431942694983218637, 0,
0.76476867949431942694983218637, 1.96343175165254905925358599741, 3.02095114450699448904814605772, 4.86059228013340531983625673998, 6.23842269550892094546688358687, 7.17386994725067211127780975418, 7.957858299218419687105484165076, 9.369105439194281653412679256823, 9.859242716278973788015217426737
