| L(s) = 1 | + 9.34·2-s + 9·3-s + 55.4·4-s + 70.5·5-s + 84.1·6-s − 89.7·7-s + 218.·8-s + 81·9-s + 659.·10-s − 436.·11-s + 498.·12-s − 258.·13-s − 838.·14-s + 634.·15-s + 272.·16-s + 591.·17-s + 757.·18-s + 1.66e3·19-s + 3.90e3·20-s − 807.·21-s − 4.08e3·22-s − 529·23-s + 1.96e3·24-s + 1.84e3·25-s − 2.42e3·26-s + 729·27-s − 4.97e3·28-s + ⋯ |
| L(s) = 1 | + 1.65·2-s + 0.577·3-s + 1.73·4-s + 1.26·5-s + 0.954·6-s − 0.692·7-s + 1.20·8-s + 0.333·9-s + 2.08·10-s − 1.08·11-s + 0.999·12-s − 0.424·13-s − 1.14·14-s + 0.728·15-s + 0.266·16-s + 0.496·17-s + 0.550·18-s + 1.05·19-s + 2.18·20-s − 0.399·21-s − 1.79·22-s − 0.208·23-s + 0.697·24-s + 0.591·25-s − 0.702·26-s + 0.192·27-s − 1.19·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.390357468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.390357468\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 23 | \( 1 + 529T \) |
| good | 2 | \( 1 - 9.34T + 32T^{2} \) |
| 5 | \( 1 - 70.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 89.7T + 1.68e4T^{2} \) |
| 11 | \( 1 + 436.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 258.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 591.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.66e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 5.12e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.87e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 884.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.84e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.23e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.38e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.45e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.77e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.53e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.11e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.56e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.89e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.14e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.12e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.37e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.16e4T + 8.58e9T^{2} \) |
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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
−13.57158285760811642514662672805, −13.14887498663154159641709506206, −12.06807647091299501124716761597, −10.39789656572581414984241935170, −9.398818162127121915183334138429, −7.41716477950073777319290556333, −6.02342953779757556066240162203, −5.11230797549031381038360408317, −3.33346025282642208665222941230, −2.24389398766611432919762071086,
2.24389398766611432919762071086, 3.33346025282642208665222941230, 5.11230797549031381038360408317, 6.02342953779757556066240162203, 7.41716477950073777319290556333, 9.398818162127121915183334138429, 10.39789656572581414984241935170, 12.06807647091299501124716761597, 13.14887498663154159641709506206, 13.57158285760811642514662672805
