L(s) = 1 | + 1.87·2-s − 3-s + 1.50·4-s − 0.347·5-s − 1.87·6-s + 7-s − 0.933·8-s + 9-s − 0.649·10-s + 2.00·11-s − 1.50·12-s + 4.04·13-s + 1.87·14-s + 0.347·15-s − 4.74·16-s − 1.51·17-s + 1.87·18-s − 3.13·19-s − 0.521·20-s − 21-s + 3.75·22-s − 8.53·23-s + 0.933·24-s − 4.87·25-s + 7.57·26-s − 27-s + 1.50·28-s + ⋯ |
L(s) = 1 | + 1.32·2-s − 0.577·3-s + 0.750·4-s − 0.155·5-s − 0.763·6-s + 0.377·7-s − 0.330·8-s + 0.333·9-s − 0.205·10-s + 0.605·11-s − 0.433·12-s + 1.12·13-s + 0.500·14-s + 0.0896·15-s − 1.18·16-s − 0.368·17-s + 0.441·18-s − 0.719·19-s − 0.116·20-s − 0.218·21-s + 0.801·22-s − 1.77·23-s + 0.190·24-s − 0.975·25-s + 1.48·26-s − 0.192·27-s + 0.283·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 - T \) |
good | 2 | \( 1 - 1.87T + 2T^{2} \) |
| 5 | \( 1 + 0.347T + 5T^{2} \) |
| 11 | \( 1 - 2.00T + 11T^{2} \) |
| 13 | \( 1 - 4.04T + 13T^{2} \) |
| 17 | \( 1 + 1.51T + 17T^{2} \) |
| 19 | \( 1 + 3.13T + 19T^{2} \) |
| 23 | \( 1 + 8.53T + 23T^{2} \) |
| 29 | \( 1 - 7.74T + 29T^{2} \) |
| 31 | \( 1 - 0.247T + 31T^{2} \) |
| 37 | \( 1 + 7.98T + 37T^{2} \) |
| 41 | \( 1 - 2.53T + 41T^{2} \) |
| 43 | \( 1 - 6.13T + 43T^{2} \) |
| 47 | \( 1 + 5.03T + 47T^{2} \) |
| 53 | \( 1 - 6.46T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 3.84T + 61T^{2} \) |
| 67 | \( 1 - 3.88T + 67T^{2} \) |
| 71 | \( 1 + 0.726T + 71T^{2} \) |
| 73 | \( 1 + 3.20T + 73T^{2} \) |
| 79 | \( 1 + 6.99T + 79T^{2} \) |
| 83 | \( 1 - 8.12T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−7.19815002558356029679689221493, −6.30729630887370438430892113881, −6.14627804276199666573475238938, −5.39733793207509064265438044579, −4.47137681829673269144901938252, −4.12927676967799359997402587270, −3.48236007570432852604284290926, −2.36356767547905551536936454753, −1.44887682091787935263573968334, 0,
1.44887682091787935263573968334, 2.36356767547905551536936454753, 3.48236007570432852604284290926, 4.12927676967799359997402587270, 4.47137681829673269144901938252, 5.39733793207509064265438044579, 6.14627804276199666573475238938, 6.30729630887370438430892113881, 7.19815002558356029679689221493