| L(s) = 1 | + 2.34e6·2-s − 1.04e10·3-s − 3.30e12·4-s − 6.18e14·5-s − 2.45e16·6-s − 6.01e16·7-s − 2.83e19·8-s + 1.09e20·9-s − 1.45e21·10-s − 9.76e21·11-s + 3.45e22·12-s − 7.58e23·13-s − 1.41e23·14-s + 6.47e24·15-s − 3.74e25·16-s + 2.78e26·17-s + 2.56e26·18-s + 4.05e27·19-s + 2.04e27·20-s + 6.29e26·21-s − 2.28e28·22-s + 1.45e29·23-s + 2.96e29·24-s − 7.53e29·25-s − 1.77e30·26-s − 1.14e30·27-s + 1.98e29·28-s + ⋯ |
| L(s) = 1 | + 0.790·2-s − 0.577·3-s − 0.375·4-s − 0.580·5-s − 0.456·6-s − 0.0407·7-s − 1.08·8-s + 0.333·9-s − 0.458·10-s − 0.397·11-s + 0.216·12-s − 0.851·13-s − 0.0321·14-s + 0.335·15-s − 0.484·16-s + 0.977·17-s + 0.263·18-s + 1.30·19-s + 0.217·20-s + 0.0234·21-s − 0.314·22-s + 0.770·23-s + 0.627·24-s − 0.663·25-s − 0.672·26-s − 0.192·27-s + 0.0152·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(22)\) |
\(\approx\) |
\(1.436854381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.436854381\) |
| \(L(\frac{45}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 1.04e10T \) |
| good | 2 | \( 1 - 2.34e6T + 8.79e12T^{2} \) |
| 5 | \( 1 + 6.18e14T + 1.13e30T^{2} \) |
| 7 | \( 1 + 6.01e16T + 2.18e36T^{2} \) |
| 11 | \( 1 + 9.76e21T + 6.02e44T^{2} \) |
| 13 | \( 1 + 7.58e23T + 7.93e47T^{2} \) |
| 17 | \( 1 - 2.78e26T + 8.11e52T^{2} \) |
| 19 | \( 1 - 4.05e27T + 9.69e54T^{2} \) |
| 23 | \( 1 - 1.45e29T + 3.58e58T^{2} \) |
| 29 | \( 1 + 9.74e30T + 7.64e62T^{2} \) |
| 31 | \( 1 - 1.20e32T + 1.34e64T^{2} \) |
| 37 | \( 1 - 9.29e32T + 2.70e67T^{2} \) |
| 41 | \( 1 - 7.39e34T + 2.23e69T^{2} \) |
| 43 | \( 1 - 1.90e35T + 1.73e70T^{2} \) |
| 47 | \( 1 + 3.68e35T + 7.94e71T^{2} \) |
| 53 | \( 1 + 1.07e37T + 1.39e74T^{2} \) |
| 59 | \( 1 + 1.44e38T + 1.40e76T^{2} \) |
| 61 | \( 1 - 4.49e38T + 5.87e76T^{2} \) |
| 67 | \( 1 - 5.03e38T + 3.32e78T^{2} \) |
| 71 | \( 1 + 4.05e39T + 4.01e79T^{2} \) |
| 73 | \( 1 + 1.07e40T + 1.32e80T^{2} \) |
| 79 | \( 1 + 8.21e40T + 3.96e81T^{2} \) |
| 83 | \( 1 - 4.07e40T + 3.31e82T^{2} \) |
| 89 | \( 1 - 1.83e41T + 6.66e83T^{2} \) |
| 97 | \( 1 + 2.07e42T + 2.69e85T^{2} \) |
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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
−15.83866274792106010593804288637, −14.35613641564986718144226269609, −12.76183149110539759669823484079, −11.64309559002309743262080447413, −9.670431316537512464333217920128, −7.61861110999939662093642845582, −5.69270121533482003648502015691, −4.55899964138300534368732447879, −3.09343388892571771053287590148, −0.66394332846954609570244219442,
0.66394332846954609570244219442, 3.09343388892571771053287590148, 4.55899964138300534368732447879, 5.69270121533482003648502015691, 7.61861110999939662093642845582, 9.670431316537512464333217920128, 11.64309559002309743262080447413, 12.76183149110539759669823484079, 14.35613641564986718144226269609, 15.83866274792106010593804288637
