| L(s) = 1 | − 9·3-s − 37.4·5-s − 154.·7-s + 81·9-s − 521.·11-s − 28.8·13-s + 337.·15-s − 428.·17-s + 2.56e3·19-s + 1.39e3·21-s + 529·23-s − 1.71e3·25-s − 729·27-s − 2.40e3·29-s + 2.13e3·31-s + 4.69e3·33-s + 5.80e3·35-s + 3.65e3·37-s + 259.·39-s + 1.61e4·41-s − 6.50e3·43-s − 3.03e3·45-s + 2.07e4·47-s + 7.17e3·49-s + 3.85e3·51-s + 3.43e4·53-s + 1.95e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.670·5-s − 1.19·7-s + 0.333·9-s − 1.29·11-s − 0.0473·13-s + 0.387·15-s − 0.359·17-s + 1.62·19-s + 0.689·21-s + 0.208·23-s − 0.550·25-s − 0.192·27-s − 0.530·29-s + 0.398·31-s + 0.749·33-s + 0.801·35-s + 0.438·37-s + 0.0273·39-s + 1.50·41-s − 0.536·43-s − 0.223·45-s + 1.36·47-s + 0.426·49-s + 0.207·51-s + 1.68·53-s + 0.871·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 23 | \( 1 - 529T \) |
| good | 5 | \( 1 + 37.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 154.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 521.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 28.8T + 3.71e5T^{2} \) |
| 17 | \( 1 + 428.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.56e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 2.40e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.13e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.65e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.61e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.50e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.07e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.43e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.82e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.65e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.63e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.94e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.74e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.73e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.69e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.29e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.14e4T + 8.58e9T^{2} \) |
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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
−8.757484488595469110480385850443, −7.56334736530587264912086737354, −7.26237422901096437831982404640, −6.04484175148750312128601306349, −5.43248953470614135121227645654, −4.35156559788810810283549248792, −3.37005263596745573663186867036, −2.50195451588564472493917493332, −0.827127904160252120404506073615, 0,
0.827127904160252120404506073615, 2.50195451588564472493917493332, 3.37005263596745573663186867036, 4.35156559788810810283549248792, 5.43248953470614135121227645654, 6.04484175148750312128601306349, 7.26237422901096437831982404640, 7.56334736530587264912086737354, 8.757484488595469110480385850443
