| L(s) = 1 | + 10.4·2-s − 30.2·3-s + 76.7·4-s − 315.·6-s + 254.·7-s + 467.·8-s + 670.·9-s + 51.5·11-s − 2.32e3·12-s − 255.·13-s + 2.64e3·14-s + 2.41e3·16-s + 289·17-s + 6.99e3·18-s − 1.64e3·19-s − 7.67e3·21-s + 537.·22-s − 84.1·23-s − 1.41e4·24-s − 2.66e3·26-s − 1.29e4·27-s + 1.95e4·28-s + 3.46e3·29-s + 6.83e3·31-s + 1.02e4·32-s − 1.55e3·33-s + 3.01e3·34-s + ⋯ |
| L(s) = 1 | + 1.84·2-s − 1.93·3-s + 2.39·4-s − 3.57·6-s + 1.95·7-s + 2.57·8-s + 2.75·9-s + 0.128·11-s − 4.65·12-s − 0.420·13-s + 3.61·14-s + 2.35·16-s + 0.242·17-s + 5.08·18-s − 1.04·19-s − 3.79·21-s + 0.236·22-s − 0.0331·23-s − 5.00·24-s − 0.774·26-s − 3.41·27-s + 4.70·28-s + 0.764·29-s + 1.27·31-s + 1.76·32-s − 0.249·33-s + 0.447·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.102223891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.102223891\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 - 289T \) |
| good | 2 | \( 1 - 10.4T + 32T^{2} \) |
| 3 | \( 1 + 30.2T + 243T^{2} \) |
| 7 | \( 1 - 254.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 51.5T + 1.61e5T^{2} \) |
| 13 | \( 1 + 255.T + 3.71e5T^{2} \) |
| 19 | \( 1 + 1.64e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 84.1T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.46e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.83e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.38e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.91e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.64e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 8.39e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.82e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.32e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.94e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.14e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.22e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.32e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.16e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.27e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.83e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.71e4T + 8.58e9T^{2} \) |
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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
−10.96453769642854458567118027538, −10.23370163360367366026205685994, −8.018195181079243543954067317658, −7.01723705006414202693995236386, −6.18378193683268380972436388376, −5.34311400371357324153686039850, −4.69045744866416299487341671316, −4.21547853494953188028205492382, −2.13951368141901098299294824823, −1.07159649754796867997677804315,
1.07159649754796867997677804315, 2.13951368141901098299294824823, 4.21547853494953188028205492382, 4.69045744866416299487341671316, 5.34311400371357324153686039850, 6.18378193683268380972436388376, 7.01723705006414202693995236386, 8.018195181079243543954067317658, 10.23370163360367366026205685994, 10.96453769642854458567118027538
