| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 5·5-s − 6·6-s − 11.0·7-s − 8·8-s + 9·9-s − 10·10-s − 49.3·11-s + 12·12-s + 19.1·13-s + 22.0·14-s + 15·15-s + 16·16-s + 40.5·17-s − 18·18-s − 36.3·19-s + 20·20-s − 33.1·21-s + 98.7·22-s + 166.·23-s − 24·24-s + 25·25-s − 38.2·26-s + 27·27-s − 44.1·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.596·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 1.35·11-s + 0.288·12-s + 0.408·13-s + 0.421·14-s + 0.258·15-s + 0.250·16-s + 0.579·17-s − 0.235·18-s − 0.438·19-s + 0.223·20-s − 0.344·21-s + 0.957·22-s + 1.50·23-s − 0.204·24-s + 0.200·25-s − 0.288·26-s + 0.192·27-s − 0.298·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 31 | \( 1 - 31T \) |
| good | 7 | \( 1 + 11.0T + 343T^{2} \) |
| 11 | \( 1 + 49.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 19.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 40.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 36.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 166.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 142.T + 2.43e4T^{2} \) |
| 37 | \( 1 + 179.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 138.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 337.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 460.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 435.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 587.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 767.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 508.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 529.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 625.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 315.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 729.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 343.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 149.T + 9.12e5T^{2} \) |
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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
−9.191595334526039406283958896747, −8.570571742539865505135144791435, −7.67002756851389615973348807397, −6.92891400959662781042868945800, −5.90919138034309600960174001414, −4.95972761644890835806692215263, −3.40780850677143987434669727195, −2.67250694154389200507222626800, −1.48391043134905594734533285333, 0,
1.48391043134905594734533285333, 2.67250694154389200507222626800, 3.40780850677143987434669727195, 4.95972761644890835806692215263, 5.90919138034309600960174001414, 6.92891400959662781042868945800, 7.67002756851389615973348807397, 8.570571742539865505135144791435, 9.191595334526039406283958896747
