| L(s) = 1 | + 0.646·2-s − 3-s − 1.58·4-s − 5-s − 0.646·6-s + 7-s − 2.31·8-s + 9-s − 0.646·10-s + 5.35·11-s + 1.58·12-s − 3.19·13-s + 0.646·14-s + 15-s + 1.66·16-s − 4.61·17-s + 0.646·18-s − 3.53·19-s + 1.58·20-s − 21-s + 3.46·22-s − 23-s + 2.31·24-s + 25-s − 2.06·26-s − 27-s − 1.58·28-s + ⋯ |
| L(s) = 1 | + 0.457·2-s − 0.577·3-s − 0.790·4-s − 0.447·5-s − 0.263·6-s + 0.377·7-s − 0.818·8-s + 0.333·9-s − 0.204·10-s + 1.61·11-s + 0.456·12-s − 0.886·13-s + 0.172·14-s + 0.258·15-s + 0.416·16-s − 1.11·17-s + 0.152·18-s − 0.810·19-s + 0.353·20-s − 0.218·21-s + 0.738·22-s − 0.208·23-s + 0.472·24-s + 0.200·25-s − 0.405·26-s − 0.192·27-s − 0.298·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.159907990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.159907990\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 0.646T + 2T^{2} \) |
| 11 | \( 1 - 5.35T + 11T^{2} \) |
| 13 | \( 1 + 3.19T + 13T^{2} \) |
| 17 | \( 1 + 4.61T + 17T^{2} \) |
| 19 | \( 1 + 3.53T + 19T^{2} \) |
| 29 | \( 1 - 0.469T + 29T^{2} \) |
| 31 | \( 1 + 4.53T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 - 9.01T + 47T^{2} \) |
| 53 | \( 1 + 7.78T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 - 5.51T + 67T^{2} \) |
| 71 | \( 1 + 2.01T + 71T^{2} \) |
| 73 | \( 1 - 6.68T + 73T^{2} \) |
| 79 | \( 1 + 5.15T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 - 4.27T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−9.080472382614556421671822078156, −8.263406480179018123401997561263, −7.30414711839848643207328398401, −6.48793436370679190715180951068, −5.81879912245571890399693299100, −4.69356661504649976984989661874, −4.37167748121079573954941955894, −3.58129308924506522184811909972, −2.13339056573458886881262046587, −0.66271853547962588883432661710,
0.66271853547962588883432661710, 2.13339056573458886881262046587, 3.58129308924506522184811909972, 4.37167748121079573954941955894, 4.69356661504649976984989661874, 5.81879912245571890399693299100, 6.48793436370679190715180951068, 7.30414711839848643207328398401, 8.263406480179018123401997561263, 9.080472382614556421671822078156
