| L(s) = 1 | − 1.24·2-s − 3.40·3-s − 0.445·4-s − 5-s + 4.24·6-s + 7-s + 3.04·8-s + 8.61·9-s + 1.24·10-s − 2.47·11-s + 1.51·12-s − 1.24·14-s + 3.40·15-s − 2.91·16-s + 2.15·17-s − 10.7·18-s + 5.39·19-s + 0.445·20-s − 3.40·21-s + 3.08·22-s − 0.0780·23-s − 10.3·24-s + 25-s − 19.1·27-s − 0.445·28-s + 5.91·29-s − 4.24·30-s + ⋯ |
| L(s) = 1 | − 0.881·2-s − 1.96·3-s − 0.222·4-s − 0.447·5-s + 1.73·6-s + 0.377·7-s + 1.07·8-s + 2.87·9-s + 0.394·10-s − 0.746·11-s + 0.437·12-s − 0.333·14-s + 0.879·15-s − 0.727·16-s + 0.523·17-s − 2.53·18-s + 1.23·19-s + 0.0995·20-s − 0.743·21-s + 0.658·22-s − 0.0162·23-s − 2.12·24-s + 0.200·25-s − 3.68·27-s − 0.0841·28-s + 1.09·29-s − 0.775·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 3 | \( 1 + 3.40T + 3T^{2} \) |
| 11 | \( 1 + 2.47T + 11T^{2} \) |
| 17 | \( 1 - 2.15T + 17T^{2} \) |
| 19 | \( 1 - 5.39T + 19T^{2} \) |
| 23 | \( 1 + 0.0780T + 23T^{2} \) |
| 29 | \( 1 - 5.91T + 29T^{2} \) |
| 31 | \( 1 - 1.42T + 31T^{2} \) |
| 37 | \( 1 + 6.78T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 + 6.09T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 + 8.95T + 53T^{2} \) |
| 59 | \( 1 + 7.70T + 59T^{2} \) |
| 61 | \( 1 + 9.90T + 61T^{2} \) |
| 67 | \( 1 - 4.66T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + 0.154T + 73T^{2} \) |
| 79 | \( 1 - 3.80T + 79T^{2} \) |
| 83 | \( 1 + 5.79T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + 1.05T + 97T^{2} \) |
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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
−7.71338240576113876932031041610, −7.10705577748789168305867172194, −6.41916276669119849629906419996, −5.34291823922473509377170717167, −5.08017526910669373292092891997, −4.42295407357441533852952228409, −3.39164274505278757790255104095, −1.67564560328386574729422982768, −0.902149151470358416506667789164, 0,
0.902149151470358416506667789164, 1.67564560328386574729422982768, 3.39164274505278757790255104095, 4.42295407357441533852952228409, 5.08017526910669373292092891997, 5.34291823922473509377170717167, 6.41916276669119849629906419996, 7.10705577748789168305867172194, 7.71338240576113876932031041610
