| L(s) = 1 | − 3-s − 2·5-s − 2·7-s − 2·9-s + 11-s + 4·13-s + 2·15-s + 2·17-s + 2·21-s + 6·23-s − 25-s + 5·27-s − 2·29-s + 10·31-s − 33-s + 4·35-s + 10·37-s − 4·39-s + 9·41-s − 4·43-s + 4·45-s + 6·47-s − 3·49-s − 2·51-s − 2·55-s − 15·59-s − 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.755·7-s − 2/3·9-s + 0.301·11-s + 1.10·13-s + 0.516·15-s + 0.485·17-s + 0.436·21-s + 1.25·23-s − 1/5·25-s + 0.962·27-s − 0.371·29-s + 1.79·31-s − 0.174·33-s + 0.676·35-s + 1.64·37-s − 0.640·39-s + 1.40·41-s − 0.609·43-s + 0.596·45-s + 0.875·47-s − 3/7·49-s − 0.280·51-s − 0.269·55-s − 1.95·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−14.03630463996415, −13.60100256828065, −12.93957905003918, −12.63539552026949, −12.03626293897780, −11.50578036551013, −11.25299030414131, −10.85509711809310, −10.13107243196586, −9.672649684680525, −8.967932786135999, −8.693228011526045, −7.905047040676714, −7.699292258992609, −6.826736202186736, −6.427811503901191, −5.901909113211401, −5.558450828302225, −4.580758472188532, −4.310384281170511, −3.530711482465336, −3.035482127078537, −2.590347273687923, −1.289144639566666, −0.8242251532109166, 0,
0.8242251532109166, 1.289144639566666, 2.590347273687923, 3.035482127078537, 3.530711482465336, 4.310384281170511, 4.580758472188532, 5.558450828302225, 5.901909113211401, 6.427811503901191, 6.826736202186736, 7.699292258992609, 7.905047040676714, 8.693228011526045, 8.967932786135999, 9.672649684680525, 10.13107243196586, 10.85509711809310, 11.25299030414131, 11.50578036551013, 12.03626293897780, 12.63539552026949, 12.93957905003918, 13.60100256828065, 14.03630463996415
