| L(s) = 1 | + 4.47·2-s − 11.9·4-s − 25·5-s − 168.·7-s − 196.·8-s − 111.·10-s − 121·11-s + 290.·13-s − 756.·14-s − 499.·16-s − 623.·17-s − 398.·19-s + 298.·20-s − 541.·22-s − 3.78e3·23-s + 625·25-s + 1.30e3·26-s + 2.01e3·28-s − 4.22e3·29-s − 5.59e3·31-s + 4.06e3·32-s − 2.79e3·34-s + 4.22e3·35-s + 301.·37-s − 1.78e3·38-s + 4.92e3·40-s + 1.46e4·41-s + ⋯ |
| L(s) = 1 | + 0.791·2-s − 0.373·4-s − 0.447·5-s − 1.30·7-s − 1.08·8-s − 0.354·10-s − 0.301·11-s + 0.476·13-s − 1.03·14-s − 0.487·16-s − 0.523·17-s − 0.253·19-s + 0.166·20-s − 0.238·22-s − 1.49·23-s + 0.200·25-s + 0.377·26-s + 0.485·28-s − 0.931·29-s − 1.04·31-s + 0.701·32-s − 0.414·34-s + 0.582·35-s + 0.0361·37-s − 0.200·38-s + 0.486·40-s + 1.35·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9397048055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9397048055\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 - 4.47T + 32T^{2} \) |
| 7 | \( 1 + 168.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 290.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 623.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 398.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.78e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.22e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 301.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.46e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 151.T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.35e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.81e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.05e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.98e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.24e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.00e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 476.T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.56e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.80e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.16e4T + 8.58e9T^{2} \) |
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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
−10.08592627108862619270163812350, −9.269476613241301255521228839187, −8.464889724283013750176615172997, −7.26348860449073221158937344518, −6.19709274246448217644256634064, −5.52667564638945336117210039116, −4.12834240531880298761248803484, −3.65106973807353494712077878195, −2.44750071205673160746341924653, −0.41210886374434186777067571915,
0.41210886374434186777067571915, 2.44750071205673160746341924653, 3.65106973807353494712077878195, 4.12834240531880298761248803484, 5.52667564638945336117210039116, 6.19709274246448217644256634064, 7.26348860449073221158937344518, 8.464889724283013750176615172997, 9.269476613241301255521228839187, 10.08592627108862619270163812350
