| L(s) = 1 | + 5.48·2-s + 22.1·4-s − 14.0·5-s + 24.2·7-s + 77.5·8-s − 77.3·10-s − 3.10·11-s − 13·13-s + 133.·14-s + 248.·16-s − 43.9·17-s − 85.8·19-s − 311.·20-s − 17.0·22-s − 203.·23-s + 73.4·25-s − 71.3·26-s + 537.·28-s + 31.0·29-s − 135.·31-s + 744.·32-s − 241.·34-s − 341.·35-s + 290.·37-s − 471.·38-s − 1.09e3·40-s + 148.·41-s + ⋯ |
| L(s) = 1 | + 1.94·2-s + 2.76·4-s − 1.25·5-s + 1.31·7-s + 3.42·8-s − 2.44·10-s − 0.0851·11-s − 0.277·13-s + 2.54·14-s + 3.88·16-s − 0.626·17-s − 1.03·19-s − 3.48·20-s − 0.165·22-s − 1.84·23-s + 0.587·25-s − 0.538·26-s + 3.62·28-s + 0.198·29-s − 0.786·31-s + 4.11·32-s − 1.21·34-s − 1.65·35-s + 1.28·37-s − 2.01·38-s − 4.31·40-s + 0.566·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.474209945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.474209945\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 - 5.48T + 8T^{2} \) |
| 5 | \( 1 + 14.0T + 125T^{2} \) |
| 7 | \( 1 - 24.2T + 343T^{2} \) |
| 11 | \( 1 + 3.10T + 1.33e3T^{2} \) |
| 17 | \( 1 + 43.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 85.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 203.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 31.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 135.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 290.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 148.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 281.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 225.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 172.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 41.2T + 2.05e5T^{2} \) |
| 61 | \( 1 - 499.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 503.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 946.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.11e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 674.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 59.4T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.21e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 879.T + 9.12e5T^{2} \) |
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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
−13.01746062468504012101770449778, −12.07246391235270536303636767296, −11.43145904352010844306592673242, −10.68201468391498145500528056181, −8.140572683382447779593292704795, −7.37998279142584114963315877880, −5.95466041927721683549380325541, −4.56525955689829671607952728377, −4.00599306635169217245363092163, −2.20702796588499446205262391721,
2.20702796588499446205262391721, 4.00599306635169217245363092163, 4.56525955689829671607952728377, 5.95466041927721683549380325541, 7.37998279142584114963315877880, 8.140572683382447779593292704795, 10.68201468391498145500528056181, 11.43145904352010844306592673242, 12.07246391235270536303636767296, 13.01746062468504012101770449778
