| L(s) = 1 | + 3.84·2-s − 0.121·3-s + 6.81·4-s + 8.75·5-s − 0.468·6-s − 0.407·7-s − 4.55·8-s − 26.9·9-s + 33.7·10-s − 0.829·12-s − 13·13-s − 1.56·14-s − 1.06·15-s − 72.0·16-s + 32.1·17-s − 103.·18-s + 114.·19-s + 59.7·20-s + 0.0496·21-s − 119.·23-s + 0.554·24-s − 48.2·25-s − 50.0·26-s + 6.57·27-s − 2.77·28-s − 31.5·29-s − 4.10·30-s + ⋯ |
| L(s) = 1 | + 1.36·2-s − 0.0234·3-s + 0.851·4-s + 0.783·5-s − 0.0318·6-s − 0.0220·7-s − 0.201·8-s − 0.999·9-s + 1.06·10-s − 0.0199·12-s − 0.277·13-s − 0.0299·14-s − 0.0183·15-s − 1.12·16-s + 0.459·17-s − 1.36·18-s + 1.38·19-s + 0.667·20-s + 0.000515·21-s − 1.08·23-s + 0.00472·24-s − 0.386·25-s − 0.377·26-s + 0.0468·27-s − 0.0187·28-s − 0.202·29-s − 0.0249·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 - 3.84T + 8T^{2} \) |
| 3 | \( 1 + 0.121T + 27T^{2} \) |
| 5 | \( 1 - 8.75T + 125T^{2} \) |
| 7 | \( 1 + 0.407T + 343T^{2} \) |
| 17 | \( 1 - 32.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 114.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 119.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 31.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 85.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 60.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 108.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 409.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 507.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 64.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 652.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 724.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 396.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 51.0T + 3.57e5T^{2} \) |
| 73 | \( 1 + 372.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 736.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 836.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.15e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.59e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.663448392061399859662587596168, −7.76153991121387051024556906390, −6.66024794584713318588725678576, −5.86163822936268470866119380313, −5.45107836773309240795165791462, −4.61435769975412848368894129592, −3.45004121314141424863210614356, −2.83391430478392953289867727225, −1.72060578950474803269214069331, 0,
1.72060578950474803269214069331, 2.83391430478392953289867727225, 3.45004121314141424863210614356, 4.61435769975412848368894129592, 5.45107836773309240795165791462, 5.86163822936268470866119380313, 6.66024794584713318588725678576, 7.76153991121387051024556906390, 8.663448392061399859662587596168
