L(s) = 1 | − 2-s − 3-s + 4-s + 1.34·5-s + 6-s − 3.24·7-s − 8-s + 9-s − 1.34·10-s − 3.77·11-s − 12-s + 13-s + 3.24·14-s − 1.34·15-s + 16-s − 7.24·17-s − 18-s + 0.661·19-s + 1.34·20-s + 3.24·21-s + 3.77·22-s − 3.88·23-s + 24-s − 3.18·25-s − 26-s − 27-s − 3.24·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.601·5-s + 0.408·6-s − 1.22·7-s − 0.353·8-s + 0.333·9-s − 0.425·10-s − 1.13·11-s − 0.288·12-s + 0.277·13-s + 0.867·14-s − 0.347·15-s + 0.250·16-s − 1.75·17-s − 0.235·18-s + 0.151·19-s + 0.300·20-s + 0.708·21-s + 0.804·22-s − 0.810·23-s + 0.204·24-s − 0.637·25-s − 0.196·26-s − 0.192·27-s − 0.613·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3102099440\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3102099440\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 - 1.34T + 5T^{2} \) |
| 7 | \( 1 + 3.24T + 7T^{2} \) |
| 11 | \( 1 + 3.77T + 11T^{2} \) |
| 17 | \( 1 + 7.24T + 17T^{2} \) |
| 19 | \( 1 - 0.661T + 19T^{2} \) |
| 23 | \( 1 + 3.88T + 23T^{2} \) |
| 29 | \( 1 + 2.76T + 29T^{2} \) |
| 31 | \( 1 - 7.79T + 31T^{2} \) |
| 37 | \( 1 + 5.09T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 - 0.261T + 43T^{2} \) |
| 47 | \( 1 - 0.430T + 47T^{2} \) |
| 53 | \( 1 + 6.98T + 53T^{2} \) |
| 59 | \( 1 - 5.79T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 15.9T + 67T^{2} \) |
| 71 | \( 1 - 1.48T + 71T^{2} \) |
| 73 | \( 1 + 7.24T + 73T^{2} \) |
| 79 | \( 1 - 9.27T + 79T^{2} \) |
| 83 | \( 1 - 3.12T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 + 4.13T + 97T^{2} \) |
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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
−7.923375835640141658369531847093, −6.88800478824989784321769940072, −6.59224711959938823030230250600, −5.91321135100916864873610351141, −5.26949977481949733748276227255, −4.32723625323309499244398871124, −3.34134512121663744495903923486, −2.47771682343444644848139151165, −1.75847756131711736160238542532, −0.29849432704311060972946178695,
0.29849432704311060972946178695, 1.75847756131711736160238542532, 2.47771682343444644848139151165, 3.34134512121663744495903923486, 4.32723625323309499244398871124, 5.26949977481949733748276227255, 5.91321135100916864873610351141, 6.59224711959938823030230250600, 6.88800478824989784321769940072, 7.923375835640141658369531847093