| L(s) = 1 | − 1.01·2-s − 0.976·4-s − 1.54·5-s + 4.83·7-s + 3.01·8-s + 1.55·10-s − 4.58·11-s + 0.937·13-s − 4.89·14-s − 1.09·16-s − 5.94·17-s + 1.55·19-s + 1.50·20-s + 4.63·22-s + 3.42·23-s − 2.62·25-s − 0.948·26-s − 4.71·28-s − 5.41·29-s − 0.550·31-s − 4.91·32-s + 6.01·34-s − 7.44·35-s + 3.01·37-s − 1.57·38-s − 4.63·40-s + 1.56·41-s + ⋯ |
| L(s) = 1 | − 0.715·2-s − 0.488·4-s − 0.688·5-s + 1.82·7-s + 1.06·8-s + 0.492·10-s − 1.38·11-s + 0.260·13-s − 1.30·14-s − 0.273·16-s − 1.44·17-s + 0.357·19-s + 0.336·20-s + 0.989·22-s + 0.714·23-s − 0.525·25-s − 0.186·26-s − 0.891·28-s − 1.00·29-s − 0.0989·31-s − 0.868·32-s + 1.03·34-s − 1.25·35-s + 0.496·37-s − 0.256·38-s − 0.733·40-s + 0.244·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8878368888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8878368888\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 2 | \( 1 + 1.01T + 2T^{2} \) |
| 5 | \( 1 + 1.54T + 5T^{2} \) |
| 7 | \( 1 - 4.83T + 7T^{2} \) |
| 11 | \( 1 + 4.58T + 11T^{2} \) |
| 13 | \( 1 - 0.937T + 13T^{2} \) |
| 17 | \( 1 + 5.94T + 17T^{2} \) |
| 19 | \( 1 - 1.55T + 19T^{2} \) |
| 23 | \( 1 - 3.42T + 23T^{2} \) |
| 29 | \( 1 + 5.41T + 29T^{2} \) |
| 31 | \( 1 + 0.550T + 31T^{2} \) |
| 37 | \( 1 - 3.01T + 37T^{2} \) |
| 41 | \( 1 - 1.56T + 41T^{2} \) |
| 43 | \( 1 - 4.62T + 43T^{2} \) |
| 47 | \( 1 + 2.13T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 - 6.96T + 59T^{2} \) |
| 61 | \( 1 + 6.57T + 61T^{2} \) |
| 67 | \( 1 - 6.39T + 67T^{2} \) |
| 71 | \( 1 - 4.75T + 71T^{2} \) |
| 73 | \( 1 + 9.95T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 9.08T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−8.167472348896933266003845258215, −7.53458708294332797319115608806, −7.20261368847148262024021295404, −5.71694293559291192957850045368, −5.09174532473146670786984342124, −4.48246410509039362405676910505, −3.90660695735065895784428091128, −2.51482844936199698619345046288, −1.68792499243584833624495616031, −0.56699535611783611469009132375,
0.56699535611783611469009132375, 1.68792499243584833624495616031, 2.51482844936199698619345046288, 3.90660695735065895784428091128, 4.48246410509039362405676910505, 5.09174532473146670786984342124, 5.71694293559291192957850045368, 7.20261368847148262024021295404, 7.53458708294332797319115608806, 8.167472348896933266003845258215
