L(s) = 1 | + 0.921·2-s − 1.15·4-s − 1.07·5-s + 3.99·7-s − 2.90·8-s − 0.991·10-s − 3.85·11-s + 1.10·13-s + 3.68·14-s − 0.370·16-s − 4.98·17-s + 6.77·19-s + 1.24·20-s − 3.55·22-s − 4.48·23-s − 3.84·25-s + 1.01·26-s − 4.60·28-s + 0.455·29-s − 0.141·31-s + 5.46·32-s − 4.58·34-s − 4.30·35-s − 5.62·37-s + 6.23·38-s + 3.12·40-s − 3.87·41-s + ⋯ |
L(s) = 1 | + 0.651·2-s − 0.575·4-s − 0.481·5-s + 1.51·7-s − 1.02·8-s − 0.313·10-s − 1.16·11-s + 0.306·13-s + 0.984·14-s − 0.0925·16-s − 1.20·17-s + 1.55·19-s + 0.277·20-s − 0.757·22-s − 0.935·23-s − 0.768·25-s + 0.199·26-s − 0.870·28-s + 0.0846·29-s − 0.0254·31-s + 0.966·32-s − 0.786·34-s − 0.727·35-s − 0.925·37-s + 1.01·38-s + 0.494·40-s − 0.605·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 239 | \( 1 + T \) |
good | 2 | \( 1 - 0.921T + 2T^{2} \) |
| 5 | \( 1 + 1.07T + 5T^{2} \) |
| 7 | \( 1 - 3.99T + 7T^{2} \) |
| 11 | \( 1 + 3.85T + 11T^{2} \) |
| 13 | \( 1 - 1.10T + 13T^{2} \) |
| 17 | \( 1 + 4.98T + 17T^{2} \) |
| 19 | \( 1 - 6.77T + 19T^{2} \) |
| 23 | \( 1 + 4.48T + 23T^{2} \) |
| 29 | \( 1 - 0.455T + 29T^{2} \) |
| 31 | \( 1 + 0.141T + 31T^{2} \) |
| 37 | \( 1 + 5.62T + 37T^{2} \) |
| 41 | \( 1 + 3.87T + 41T^{2} \) |
| 43 | \( 1 + 5.21T + 43T^{2} \) |
| 47 | \( 1 - 0.0217T + 47T^{2} \) |
| 53 | \( 1 - 8.05T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 + 8.40T + 67T^{2} \) |
| 71 | \( 1 + 1.78T + 71T^{2} \) |
| 73 | \( 1 - 8.35T + 73T^{2} \) |
| 79 | \( 1 + 16.7T + 79T^{2} \) |
| 83 | \( 1 - 3.41T + 83T^{2} \) |
| 89 | \( 1 - 7.49T + 89T^{2} \) |
| 97 | \( 1 + 9.46T + 97T^{2} \) |
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
−8.511623714018949411732014570405, −7.983352797847098788877848629463, −7.33004512116084406241202762018, −6.02977912775314505547746515309, −5.23421106485419568689792399012, −4.72665230419124240934630771206, −3.94993613842333357940492918179, −2.94075709323790718664743018462, −1.68871384069805730792205188197, 0,
1.68871384069805730792205188197, 2.94075709323790718664743018462, 3.94993613842333357940492918179, 4.72665230419124240934630771206, 5.23421106485419568689792399012, 6.02977912775314505547746515309, 7.33004512116084406241202762018, 7.983352797847098788877848629463, 8.511623714018949411732014570405