| L(s) = 1 | − 3-s − 4.92·7-s + 9-s − 1.38·11-s + 13-s + 0.195·17-s + 4.92·21-s − 2.19·23-s − 27-s + 7.49·29-s + 1.38·33-s + 6.05·37-s − 39-s − 2.11·41-s + 3.49·43-s + 2.31·47-s + 17.2·49-s − 0.195·51-s − 6.05·53-s + 9.43·59-s − 3.69·61-s − 4.92·63-s − 12.6·67-s + 2.19·69-s + 13.9·71-s + 4.73·73-s + 6.81·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.86·7-s + 0.333·9-s − 0.417·11-s + 0.277·13-s + 0.0473·17-s + 1.07·21-s − 0.457·23-s − 0.192·27-s + 1.39·29-s + 0.240·33-s + 0.994·37-s − 0.160·39-s − 0.330·41-s + 0.533·43-s + 0.337·47-s + 2.46·49-s − 0.0273·51-s − 0.831·53-s + 1.22·59-s − 0.473·61-s − 0.620·63-s − 1.54·67-s + 0.264·69-s + 1.65·71-s + 0.553·73-s + 0.776·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4.92T + 7T^{2} \) |
| 11 | \( 1 + 1.38T + 11T^{2} \) |
| 17 | \( 1 - 0.195T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 - 7.49T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6.05T + 37T^{2} \) |
| 41 | \( 1 + 2.11T + 41T^{2} \) |
| 43 | \( 1 - 3.49T + 43T^{2} \) |
| 47 | \( 1 - 2.31T + 47T^{2} \) |
| 53 | \( 1 + 6.05T + 53T^{2} \) |
| 59 | \( 1 - 9.43T + 59T^{2} \) |
| 61 | \( 1 + 3.69T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 - 4.73T + 73T^{2} \) |
| 79 | \( 1 + 8.07T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 3.02T + 89T^{2} \) |
| 97 | \( 1 - 6.01T + 97T^{2} \) |
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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
−7.34569535517320693451835958715, −6.65695546191873197152690726250, −6.17072870547805019946313326565, −5.63492599930712731556908024085, −4.67373987725238618088019300888, −3.89014396352738043094274084762, −3.11434351116710432370704701040, −2.41380220245548111843668225323, −0.982944537878363355669773880261, 0,
0.982944537878363355669773880261, 2.41380220245548111843668225323, 3.11434351116710432370704701040, 3.89014396352738043094274084762, 4.67373987725238618088019300888, 5.63492599930712731556908024085, 6.17072870547805019946313326565, 6.65695546191873197152690726250, 7.34569535517320693451835958715
