L(s) = 1 | − 0.726·2-s − 3-s − 1.47·4-s + 0.449·5-s + 0.726·6-s + 4.04·7-s + 2.52·8-s + 9-s − 0.326·10-s − 1.83·11-s + 1.47·12-s − 13-s − 2.93·14-s − 0.449·15-s + 1.11·16-s − 4.56·17-s − 0.726·18-s + 6.19·19-s − 0.661·20-s − 4.04·21-s + 1.33·22-s − 0.193·23-s − 2.52·24-s − 4.79·25-s + 0.726·26-s − 27-s − 5.96·28-s + ⋯ |
L(s) = 1 | − 0.513·2-s − 0.577·3-s − 0.736·4-s + 0.201·5-s + 0.296·6-s + 1.52·7-s + 0.891·8-s + 0.333·9-s − 0.103·10-s − 0.553·11-s + 0.425·12-s − 0.277·13-s − 0.785·14-s − 0.116·15-s + 0.278·16-s − 1.10·17-s − 0.171·18-s + 1.42·19-s − 0.148·20-s − 0.883·21-s + 0.284·22-s − 0.0403·23-s − 0.514·24-s − 0.959·25-s + 0.142·26-s − 0.192·27-s − 1.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 + 0.726T + 2T^{2} \) |
| 5 | \( 1 - 0.449T + 5T^{2} \) |
| 7 | \( 1 - 4.04T + 7T^{2} \) |
| 11 | \( 1 + 1.83T + 11T^{2} \) |
| 17 | \( 1 + 4.56T + 17T^{2} \) |
| 19 | \( 1 - 6.19T + 19T^{2} \) |
| 23 | \( 1 + 0.193T + 23T^{2} \) |
| 29 | \( 1 + 0.219T + 29T^{2} \) |
| 31 | \( 1 - 3.13T + 31T^{2} \) |
| 37 | \( 1 + 6.01T + 37T^{2} \) |
| 41 | \( 1 + 7.72T + 41T^{2} \) |
| 43 | \( 1 + 5.53T + 43T^{2} \) |
| 47 | \( 1 + 3.39T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 0.305T + 59T^{2} \) |
| 61 | \( 1 - 5.70T + 61T^{2} \) |
| 67 | \( 1 - 9.16T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 2.59T + 73T^{2} \) |
| 79 | \( 1 + 7.52T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 8.73T + 89T^{2} \) |
| 97 | \( 1 - 5.76T + 97T^{2} \) |
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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
−8.149645746610983869710035417041, −7.55265502123057581252655007831, −6.76071673899868750553826387231, −5.58667246640471834313082789006, −4.99838324208810040629880499055, −4.63226769190156397520645122841, −3.55051516433569493228975263287, −2.08020490739730692506369874533, −1.29913105263282175249026392762, 0,
1.29913105263282175249026392762, 2.08020490739730692506369874533, 3.55051516433569493228975263287, 4.63226769190156397520645122841, 4.99838324208810040629880499055, 5.58667246640471834313082789006, 6.76071673899868750553826387231, 7.55265502123057581252655007831, 8.149645746610983869710035417041