L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 5.21·5-s + 6·6-s + 31.4·7-s + 8·8-s + 9·9-s − 10.4·10-s − 21.2·11-s + 12·12-s − 56.2·13-s + 62.9·14-s − 15.6·15-s + 16·16-s + 17.2·17-s + 18·18-s − 20.8·20-s + 94.4·21-s − 42.4·22-s − 206.·23-s + 24·24-s − 97.8·25-s − 112.·26-s + 27·27-s + 125.·28-s − 206.·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.466·5-s + 0.408·6-s + 1.70·7-s + 0.353·8-s + 0.333·9-s − 0.329·10-s − 0.581·11-s + 0.288·12-s − 1.20·13-s + 1.20·14-s − 0.269·15-s + 0.250·16-s + 0.246·17-s + 0.235·18-s − 0.233·20-s + 0.981·21-s − 0.411·22-s − 1.87·23-s + 0.204·24-s − 0.782·25-s − 0.848·26-s + 0.192·27-s + 0.850·28-s − 1.32·29-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 | \( 1 + 5.21T + 125T^{2} \) |
| 7 | \( 1 - 31.4T + 343T^{2} \) |
| 11 | \( 1 + 21.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 56.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 17.2T + 4.91e3T^{2} \) |
| 23 | \( 1 + 206.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 206.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 129.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 440.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 435.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 129.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 108.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 407.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 116.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 340.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 210.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 158.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 573.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 885.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 573.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 215.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 528.T + 9.12e5T^{2} \) |
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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
−7.989650497026993419497424191176, −7.72733973931323586713867158302, −7.00858391644437634667392949510, −5.57506275932175800722478683524, −5.12109511745213571704598672463, −4.21449279135148404476866594604, −3.55080273956737789592883180762, −2.18665981137565323238422930099, −1.80203237118031250041125231170, 0,
1.80203237118031250041125231170, 2.18665981137565323238422930099, 3.55080273956737789592883180762, 4.21449279135148404476866594604, 5.12109511745213571704598672463, 5.57506275932175800722478683524, 7.00858391644437634667392949510, 7.72733973931323586713867158302, 7.989650497026993419497424191176