| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 0.909·5-s + 6·6-s − 16.3·7-s + 8·8-s + 9·9-s − 1.81·10-s − 1.94·11-s + 12·12-s + 15.0·13-s − 32.6·14-s − 2.72·15-s + 16·16-s + 32.0·17-s + 18·18-s − 3.63·20-s − 48.9·21-s − 3.88·22-s − 44.8·23-s + 24·24-s − 124.·25-s + 30.1·26-s + 27·27-s − 65.3·28-s + 31.1·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.0813·5-s + 0.408·6-s − 0.881·7-s + 0.353·8-s + 0.333·9-s − 0.0575·10-s − 0.0532·11-s + 0.288·12-s + 0.321·13-s − 0.623·14-s − 0.0469·15-s + 0.250·16-s + 0.456·17-s + 0.235·18-s − 0.0406·20-s − 0.508·21-s − 0.0376·22-s − 0.406·23-s + 0.204·24-s − 0.993·25-s + 0.227·26-s + 0.192·27-s − 0.440·28-s + 0.199·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 0.909T + 125T^{2} \) |
| 7 | \( 1 + 16.3T + 343T^{2} \) |
| 11 | \( 1 + 1.94T + 1.33e3T^{2} \) |
| 13 | \( 1 - 15.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 32.0T + 4.91e3T^{2} \) |
| 23 | \( 1 + 44.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 31.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 299.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 252.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 81.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 217.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 187.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 11.8T + 1.48e5T^{2} \) |
| 59 | \( 1 - 244.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 526.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 210.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 503.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 115.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 220.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 200.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 174.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.10e3T + 9.12e5T^{2} \) |
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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
−8.263422755456844582697810102977, −7.45137071620747141844427692051, −6.75575822488754397896106522873, −5.90852378574887775809878217043, −5.17303782526881701013523669880, −3.92231868785952042072083790861, −3.54196768744657859336427383116, −2.55245501851747760710249486406, −1.54258784359168848706279111674, 0,
1.54258784359168848706279111674, 2.55245501851747760710249486406, 3.54196768744657859336427383116, 3.92231868785952042072083790861, 5.17303782526881701013523669880, 5.90852378574887775809878217043, 6.75575822488754397896106522873, 7.45137071620747141844427692051, 8.263422755456844582697810102977
