| L(s) = 1 | − 0.236·5-s − 7-s + 0.381·11-s − 5.47·13-s − 0.854·17-s + 19-s + 3.76·23-s − 4.94·25-s + 2.38·29-s + 2.85·31-s + 0.236·35-s − 3.76·37-s + 8.56·41-s − 4.47·43-s + 1.47·47-s + 49-s + 5.09·53-s − 0.0901·55-s − 11·59-s + 0.236·61-s + 1.29·65-s + 12.5·67-s + 8.23·71-s + 1.38·73-s − 0.381·77-s + 4.47·79-s + 9.56·83-s + ⋯ |
| L(s) = 1 | − 0.105·5-s − 0.377·7-s + 0.115·11-s − 1.51·13-s − 0.207·17-s + 0.229·19-s + 0.784·23-s − 0.988·25-s + 0.442·29-s + 0.512·31-s + 0.0399·35-s − 0.618·37-s + 1.33·41-s − 0.681·43-s + 0.214·47-s + 0.142·49-s + 0.699·53-s − 0.0121·55-s − 1.43·59-s + 0.0302·61-s + 0.160·65-s + 1.53·67-s + 0.977·71-s + 0.161·73-s − 0.0435·77-s + 0.503·79-s + 1.04·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.404139547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.404139547\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 0.236T + 5T^{2} \) |
| 11 | \( 1 - 0.381T + 11T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 + 0.854T + 17T^{2} \) |
| 23 | \( 1 - 3.76T + 23T^{2} \) |
| 29 | \( 1 - 2.38T + 29T^{2} \) |
| 31 | \( 1 - 2.85T + 31T^{2} \) |
| 37 | \( 1 + 3.76T + 37T^{2} \) |
| 41 | \( 1 - 8.56T + 41T^{2} \) |
| 43 | \( 1 + 4.47T + 43T^{2} \) |
| 47 | \( 1 - 1.47T + 47T^{2} \) |
| 53 | \( 1 - 5.09T + 53T^{2} \) |
| 59 | \( 1 + 11T + 59T^{2} \) |
| 61 | \( 1 - 0.236T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 8.23T + 71T^{2} \) |
| 73 | \( 1 - 1.38T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 - 9.56T + 83T^{2} \) |
| 89 | \( 1 - 2.29T + 89T^{2} \) |
| 97 | \( 1 - 5.47T + 97T^{2} \) |
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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
−8.187609483925142055726272939078, −7.52566956100109919850461449398, −6.90336896389729062104301312405, −6.17973370421833969913337415050, −5.25393581638305661136394867434, −4.65537891262678610530872460445, −3.73559602616866165646140914308, −2.83456804697838287827206118048, −2.04883608797897585665717203348, −0.63304334015876680491873545576,
0.63304334015876680491873545576, 2.04883608797897585665717203348, 2.83456804697838287827206118048, 3.73559602616866165646140914308, 4.65537891262678610530872460445, 5.25393581638305661136394867434, 6.17973370421833969913337415050, 6.90336896389729062104301312405, 7.52566956100109919850461449398, 8.187609483925142055726272939078
