| L(s) = 1 | + 9·3-s − 61.4·5-s + 46.3·7-s + 81·9-s − 16.2·11-s − 169·13-s − 552.·15-s + 659.·17-s − 17.7·19-s + 416.·21-s + 2.14e3·23-s + 646.·25-s + 729·27-s − 3.17e3·29-s − 4.03e3·31-s − 146.·33-s − 2.84e3·35-s + 1.42e4·37-s − 1.52e3·39-s − 1.78e4·41-s + 3.55e3·43-s − 4.97e3·45-s + 2.52e4·47-s − 1.46e4·49-s + 5.93e3·51-s − 5.30e3·53-s + 1.00e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.09·5-s + 0.357·7-s + 0.333·9-s − 0.0406·11-s − 0.277·13-s − 0.634·15-s + 0.553·17-s − 0.0112·19-s + 0.206·21-s + 0.843·23-s + 0.207·25-s + 0.192·27-s − 0.701·29-s − 0.753·31-s − 0.0234·33-s − 0.392·35-s + 1.71·37-s − 0.160·39-s − 1.66·41-s + 0.293·43-s − 0.366·45-s + 1.66·47-s − 0.872·49-s + 0.319·51-s − 0.259·53-s + 0.0446·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 9T \) |
| 13 | \( 1 + 169T \) |
| good | 5 | \( 1 + 61.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 46.3T + 1.68e4T^{2} \) |
| 11 | \( 1 + 16.2T + 1.61e5T^{2} \) |
| 17 | \( 1 - 659.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 17.7T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.14e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.17e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.42e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.78e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.55e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.52e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.30e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.65e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 6.51e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.11e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.10e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.91e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.99e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.72e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.97e4T + 8.58e9T^{2} \) |
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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
−9.270342274518542379589465453941, −8.431946541442639061809595164243, −7.65000389226835024268417483466, −7.09533756395803001326677995271, −5.66752325415361096522758438260, −4.54793615613510042432072461045, −3.71161525396971685437646984392, −2.72019310846678036340228897972, −1.34300963898326966150701566625, 0,
1.34300963898326966150701566625, 2.72019310846678036340228897972, 3.71161525396971685437646984392, 4.54793615613510042432072461045, 5.66752325415361096522758438260, 7.09533756395803001326677995271, 7.65000389226835024268417483466, 8.431946541442639061809595164243, 9.270342274518542379589465453941
