| L(s) = 1 | − 2·2-s + 4·4-s + 5i·5-s + 26.6·7-s − 8·8-s − 10i·10-s − 4.03i·11-s − 20.1i·13-s − 53.3·14-s + 16·16-s − 61.6i·17-s + (78.1 − 27.4i)19-s + 20i·20-s + 8.06i·22-s − 63.3i·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447i·5-s + 1.44·7-s − 0.353·8-s − 0.316i·10-s − 0.110i·11-s − 0.430i·13-s − 1.01·14-s + 0.250·16-s − 0.879i·17-s + (0.943 − 0.331i)19-s + 0.223i·20-s + 0.0781i·22-s − 0.573i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 + 0.815i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.579 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9630897564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9630897564\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5iT \) |
| 19 | \( 1 + (-78.1 + 27.4i)T \) |
| good | 7 | \( 1 - 26.6T + 343T^{2} \) |
| 11 | \( 1 + 4.03iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 20.1iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 61.6iT - 4.91e3T^{2} \) |
| 23 | \( 1 + 63.3iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 175.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 282. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 197. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 205.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 203.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 504. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 165.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 577.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 495.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 138. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 342.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 418.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 860. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 595. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 25.7T + 7.04e5T^{2} \) |
| 97 | \( 1 + 310. iT - 9.12e5T^{2} \) |
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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
−8.660543448325433369112512515231, −7.78315372514619817608397607870, −7.47271057145629518180373826648, −6.43098984997366352655408487958, −5.41761874392527961402290397762, −4.70068434989410476183160669193, −3.41386959269277344653860193925, −2.39454277413053367475379103956, −1.43058820950073919962685340156, −0.25295034375313136029143424970,
1.42928900415136775115480881662, 1.68180746768298533042224329529, 3.23330393912862029039925799273, 4.36810751734477571986427643926, 5.20654518453624742490257493972, 5.99113627870403534529716725935, 7.19658258723474063377450050530, 7.76685944444565759845897693381, 8.481235202389022289620210165847, 9.112021039438336217706736195449
