| L(s) = 1 | − 2.43·2-s + 3·3-s − 2.08·4-s − 2.10·5-s − 7.29·6-s − 2.84·7-s + 24.5·8-s + 9·9-s + 5.12·10-s − 13.8·11-s − 6.25·12-s + 25.3·13-s + 6.92·14-s − 6.31·15-s − 42.9·16-s − 81.8·17-s − 21.8·18-s − 49.3·19-s + 4.39·20-s − 8.54·21-s + 33.6·22-s + 30.0·23-s + 73.5·24-s − 120.·25-s − 61.5·26-s + 27·27-s + 5.94·28-s + ⋯ |
| L(s) = 1 | − 0.859·2-s + 0.577·3-s − 0.260·4-s − 0.188·5-s − 0.496·6-s − 0.153·7-s + 1.08·8-s + 0.333·9-s + 0.161·10-s − 0.378·11-s − 0.150·12-s + 0.540·13-s + 0.132·14-s − 0.108·15-s − 0.671·16-s − 1.16·17-s − 0.286·18-s − 0.596·19-s + 0.0490·20-s − 0.0888·21-s + 0.325·22-s + 0.272·23-s + 0.625·24-s − 0.964·25-s − 0.464·26-s + 0.192·27-s + 0.0401·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 67 | \( 1 + 67T \) |
| good | 2 | \( 1 + 2.43T + 8T^{2} \) |
| 5 | \( 1 + 2.10T + 125T^{2} \) |
| 7 | \( 1 + 2.84T + 343T^{2} \) |
| 11 | \( 1 + 13.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 25.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 81.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 49.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 30.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 188.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 334.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 346.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 86.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 219.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 167.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 193.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 336.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 258.T + 2.26e5T^{2} \) |
| 71 | \( 1 - 760.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 860.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 349.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 840.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.29e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 100.T + 9.12e5T^{2} \) |
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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
−11.15890136308483217118153707889, −10.37346033752423973582468055848, −9.306715771383547685832270359015, −8.606863924369289845870945880221, −7.77760611887880035704585595993, −6.61874301275825409927809978204, −4.89210195200042276036866401283, −3.65892642364710805721816545671, −1.86769088857789682107879958772, 0,
1.86769088857789682107879958772, 3.65892642364710805721816545671, 4.89210195200042276036866401283, 6.61874301275825409927809978204, 7.77760611887880035704585595993, 8.606863924369289845870945880221, 9.306715771383547685832270359015, 10.37346033752423973582468055848, 11.15890136308483217118153707889
