| L(s) = 1 | − 9.34·2-s + 55.4·4-s − 70.5·5-s − 89.7·7-s − 218.·8-s + 659.·10-s + 436.·11-s − 258.·13-s + 838.·14-s + 272.·16-s − 591.·17-s + 1.66e3·19-s − 3.90e3·20-s − 4.08e3·22-s + 529·23-s + 1.84e3·25-s + 2.42e3·26-s − 4.97e3·28-s + 5.12e3·29-s + 1.87e3·31-s + 4.45e3·32-s + 5.53e3·34-s + 6.32e3·35-s + 884.·37-s − 1.55e4·38-s + 1.54e4·40-s + 1.84e4·41-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.73·4-s − 1.26·5-s − 0.692·7-s − 1.20·8-s + 2.08·10-s + 1.08·11-s − 0.424·13-s + 1.14·14-s + 0.266·16-s − 0.496·17-s + 1.05·19-s − 2.18·20-s − 1.79·22-s + 0.208·23-s + 0.591·25-s + 0.702·26-s − 1.19·28-s + 1.13·29-s + 0.349·31-s + 0.768·32-s + 0.820·34-s + 0.872·35-s + 0.106·37-s − 1.74·38-s + 1.52·40-s + 1.71·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 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
−10.95117598892765534978867273016, −9.756019194797195128235835623756, −9.118169119307383881838617924732, −8.080976790296397892532181294580, −7.27959718696764679201917651309, −6.40848084673431174650897636036, −4.33688325466885155897023616713, −2.93685159139078593387133438247, −1.11475956789843685528933425306, 0,
1.11475956789843685528933425306, 2.93685159139078593387133438247, 4.33688325466885155897023616713, 6.40848084673431174650897636036, 7.27959718696764679201917651309, 8.080976790296397892532181294580, 9.118169119307383881838617924732, 9.756019194797195128235835623756, 10.95117598892765534978867273016
