| L(s) = 1 | + 4.65e6·2-s + 1.28e13·4-s − 5.54e14·5-s − 5.57e17·7-s + 1.89e19·8-s − 2.58e21·10-s + 1.13e22·11-s + 5.07e23·13-s − 2.59e24·14-s − 2.51e25·16-s − 8.73e25·17-s + 5.55e27·19-s − 7.12e27·20-s + 5.26e28·22-s − 1.85e29·23-s − 8.29e29·25-s + 2.35e30·26-s − 7.16e30·28-s − 4.19e31·29-s − 9.30e31·31-s − 2.83e32·32-s − 4.06e32·34-s + 3.09e32·35-s − 3.20e33·37-s + 2.58e34·38-s − 1.04e34·40-s − 6.48e33·41-s + ⋯ |
| L(s) = 1 | + 1.56·2-s + 1.46·4-s − 0.520·5-s − 0.377·7-s + 0.724·8-s − 0.815·10-s + 0.460·11-s + 0.569·13-s − 0.592·14-s − 0.324·16-s − 0.306·17-s + 1.78·19-s − 0.760·20-s + 0.722·22-s − 0.978·23-s − 0.729·25-s + 0.893·26-s − 0.551·28-s − 1.51·29-s − 0.802·31-s − 1.23·32-s − 0.481·34-s + 0.196·35-s − 0.616·37-s + 2.79·38-s − 0.376·40-s − 0.137·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(22)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{45}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 4.65e6T + 8.79e12T^{2} \) |
| 5 | \( 1 + 5.54e14T + 1.13e30T^{2} \) |
| 7 | \( 1 + 5.57e17T + 2.18e36T^{2} \) |
| 11 | \( 1 - 1.13e22T + 6.02e44T^{2} \) |
| 13 | \( 1 - 5.07e23T + 7.93e47T^{2} \) |
| 17 | \( 1 + 8.73e25T + 8.11e52T^{2} \) |
| 19 | \( 1 - 5.55e27T + 9.69e54T^{2} \) |
| 23 | \( 1 + 1.85e29T + 3.58e58T^{2} \) |
| 29 | \( 1 + 4.19e31T + 7.64e62T^{2} \) |
| 31 | \( 1 + 9.30e31T + 1.34e64T^{2} \) |
| 37 | \( 1 + 3.20e33T + 2.70e67T^{2} \) |
| 41 | \( 1 + 6.48e33T + 2.23e69T^{2} \) |
| 43 | \( 1 - 3.51e33T + 1.73e70T^{2} \) |
| 47 | \( 1 + 6.21e35T + 7.94e71T^{2} \) |
| 53 | \( 1 + 8.28e36T + 1.39e74T^{2} \) |
| 59 | \( 1 - 7.48e36T + 1.40e76T^{2} \) |
| 61 | \( 1 + 2.24e38T + 5.87e76T^{2} \) |
| 67 | \( 1 + 1.89e39T + 3.32e78T^{2} \) |
| 71 | \( 1 - 9.53e39T + 4.01e79T^{2} \) |
| 73 | \( 1 - 2.48e39T + 1.32e80T^{2} \) |
| 79 | \( 1 + 3.50e39T + 3.96e81T^{2} \) |
| 83 | \( 1 - 2.23e41T + 3.31e82T^{2} \) |
| 89 | \( 1 - 4.38e41T + 6.66e83T^{2} \) |
| 97 | \( 1 + 6.11e42T + 2.69e85T^{2} \) |
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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
−12.08162266250091148202152656064, −11.24560614637489468617717661007, −9.391978869723558349444080543210, −7.58977079065519591893528545125, −6.30623502381545505840460512738, −5.25806266229693142678887665710, −3.90247040411580838027297183102, −3.30263078424793757439224315892, −1.75021292791384246140093027818, 0,
1.75021292791384246140093027818, 3.30263078424793757439224315892, 3.90247040411580838027297183102, 5.25806266229693142678887665710, 6.30623502381545505840460512738, 7.58977079065519591893528545125, 9.391978869723558349444080543210, 11.24560614637489468617717661007, 12.08162266250091148202152656064
