| L(s) = 1 | + 2-s + 0.0992·3-s + 4-s − 0.713·5-s + 0.0992·6-s − 3.05·7-s + 8-s − 2.99·9-s − 0.713·10-s + 1.04·11-s + 0.0992·12-s − 2.43·13-s − 3.05·14-s − 0.0708·15-s + 16-s + 4.46·17-s − 2.99·18-s + 3.83·19-s − 0.713·20-s − 0.303·21-s + 1.04·22-s + 3.79·23-s + 0.0992·24-s − 4.49·25-s − 2.43·26-s − 0.594·27-s − 3.05·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0572·3-s + 0.5·4-s − 0.319·5-s + 0.0405·6-s − 1.15·7-s + 0.353·8-s − 0.996·9-s − 0.225·10-s + 0.315·11-s + 0.0286·12-s − 0.675·13-s − 0.817·14-s − 0.0182·15-s + 0.250·16-s + 1.08·17-s − 0.704·18-s + 0.880·19-s − 0.159·20-s − 0.0662·21-s + 0.223·22-s + 0.790·23-s + 0.0202·24-s − 0.898·25-s − 0.477·26-s − 0.114·27-s − 0.578·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.200082933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.200082933\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 - 0.0992T + 3T^{2} \) |
| 5 | \( 1 + 0.713T + 5T^{2} \) |
| 7 | \( 1 + 3.05T + 7T^{2} \) |
| 11 | \( 1 - 1.04T + 11T^{2} \) |
| 13 | \( 1 + 2.43T + 13T^{2} \) |
| 17 | \( 1 - 4.46T + 17T^{2} \) |
| 19 | \( 1 - 3.83T + 19T^{2} \) |
| 23 | \( 1 - 3.79T + 23T^{2} \) |
| 29 | \( 1 - 7.09T + 29T^{2} \) |
| 31 | \( 1 + 6.48T + 31T^{2} \) |
| 37 | \( 1 - 1.82T + 37T^{2} \) |
| 41 | \( 1 - 6.81T + 41T^{2} \) |
| 43 | \( 1 + 2.01T + 43T^{2} \) |
| 47 | \( 1 - 0.806T + 47T^{2} \) |
| 53 | \( 1 + 1.59T + 53T^{2} \) |
| 59 | \( 1 - 3.61T + 59T^{2} \) |
| 61 | \( 1 - 3.53T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 5.49T + 71T^{2} \) |
| 73 | \( 1 + 6.02T + 73T^{2} \) |
| 79 | \( 1 - 5.10T + 79T^{2} \) |
| 83 | \( 1 - 17.5T + 83T^{2} \) |
| 89 | \( 1 - 6.70T + 89T^{2} \) |
| 97 | \( 1 + 5.63T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.325006581664921583329633589196, −7.56581488499283699582945762145, −6.93024509496734734367643291828, −6.08509034639989889912233078359, −5.53365780339484787623725792957, −4.72854763159206407559430159472, −3.56268968885940776684389351243, −3.22527805390791814632053418851, −2.32981257478201164228553846694, −0.73964287905790053846823096085,
0.73964287905790053846823096085, 2.32981257478201164228553846694, 3.22527805390791814632053418851, 3.56268968885940776684389351243, 4.72854763159206407559430159472, 5.53365780339484787623725792957, 6.08509034639989889912233078359, 6.93024509496734734367643291828, 7.56581488499283699582945762145, 8.325006581664921583329633589196
