| L(s) = 1 | − 0.656·2-s − 0.204·3-s − 1.56·4-s − 1.35·5-s + 0.134·6-s + 2.34·8-s − 2.95·9-s + 0.892·10-s − 1.90·11-s + 0.321·12-s + 13-s + 0.278·15-s + 1.60·16-s + 3.56·17-s + 1.94·18-s + 0.985·19-s + 2.13·20-s + 1.25·22-s + 1.69·23-s − 0.479·24-s − 3.15·25-s − 0.656·26-s + 1.21·27-s + 6.54·29-s − 0.182·30-s + 7.69·31-s − 5.73·32-s + ⋯ |
| L(s) = 1 | − 0.463·2-s − 0.118·3-s − 0.784·4-s − 0.608·5-s + 0.0548·6-s + 0.828·8-s − 0.986·9-s + 0.282·10-s − 0.574·11-s + 0.0927·12-s + 0.277·13-s + 0.0718·15-s + 0.400·16-s + 0.864·17-s + 0.457·18-s + 0.226·19-s + 0.477·20-s + 0.266·22-s + 0.353·23-s − 0.0978·24-s − 0.630·25-s − 0.128·26-s + 0.234·27-s + 1.21·29-s − 0.0333·30-s + 1.38·31-s − 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6866795523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6866795523\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 0.656T + 2T^{2} \) |
| 3 | \( 1 + 0.204T + 3T^{2} \) |
| 5 | \( 1 + 1.35T + 5T^{2} \) |
| 11 | \( 1 + 1.90T + 11T^{2} \) |
| 17 | \( 1 - 3.56T + 17T^{2} \) |
| 19 | \( 1 - 0.985T + 19T^{2} \) |
| 23 | \( 1 - 1.69T + 23T^{2} \) |
| 29 | \( 1 - 6.54T + 29T^{2} \) |
| 31 | \( 1 - 7.69T + 31T^{2} \) |
| 37 | \( 1 + 2.02T + 37T^{2} \) |
| 41 | \( 1 - 9.88T + 41T^{2} \) |
| 43 | \( 1 + 3.16T + 43T^{2} \) |
| 47 | \( 1 - 7.76T + 47T^{2} \) |
| 53 | \( 1 - 0.354T + 53T^{2} \) |
| 59 | \( 1 - 2.16T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 9.05T + 71T^{2} \) |
| 73 | \( 1 + 7.13T + 73T^{2} \) |
| 79 | \( 1 + 5.39T + 79T^{2} \) |
| 83 | \( 1 + 2.03T + 83T^{2} \) |
| 89 | \( 1 + 6.89T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−10.43370309948356759355377065584, −9.745838140221454045526974486032, −8.633099350450562745822207012641, −8.193375193607149075719694659386, −7.34250280098881812049235129319, −5.94653348036128069079180912338, −5.07536770607284294125650802195, −4.01975644818778433237683685396, −2.84860105763173388855996623049, −0.77488385266128951688398560546,
0.77488385266128951688398560546, 2.84860105763173388855996623049, 4.01975644818778433237683685396, 5.07536770607284294125650802195, 5.94653348036128069079180912338, 7.34250280098881812049235129319, 8.193375193607149075719694659386, 8.633099350450562745822207012641, 9.745838140221454045526974486032, 10.43370309948356759355377065584
