| L(s) = 1 | − 0.306·2-s + 3-s − 1.90·4-s − 2.43·5-s − 0.306·6-s − 7-s + 1.19·8-s + 9-s + 0.746·10-s + 2.36·11-s − 1.90·12-s + 2.69·13-s + 0.306·14-s − 2.43·15-s + 3.44·16-s + 0.171·17-s − 0.306·18-s + 2.51·19-s + 4.64·20-s − 21-s − 0.723·22-s + 2.11·23-s + 1.19·24-s + 0.938·25-s − 0.824·26-s + 27-s + 1.90·28-s + ⋯ |
| L(s) = 1 | − 0.216·2-s + 0.577·3-s − 0.953·4-s − 1.08·5-s − 0.125·6-s − 0.377·7-s + 0.423·8-s + 0.333·9-s + 0.236·10-s + 0.712·11-s − 0.550·12-s + 0.746·13-s + 0.0818·14-s − 0.629·15-s + 0.861·16-s + 0.0415·17-s − 0.0722·18-s + 0.578·19-s + 1.03·20-s − 0.218·21-s − 0.154·22-s + 0.441·23-s + 0.244·24-s + 0.187·25-s − 0.161·26-s + 0.192·27-s + 0.360·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.410853838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.410853838\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 0.306T + 2T^{2} \) |
| 5 | \( 1 + 2.43T + 5T^{2} \) |
| 11 | \( 1 - 2.36T + 11T^{2} \) |
| 13 | \( 1 - 2.69T + 13T^{2} \) |
| 17 | \( 1 - 0.171T + 17T^{2} \) |
| 19 | \( 1 - 2.51T + 19T^{2} \) |
| 23 | \( 1 - 2.11T + 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 - 3.29T + 31T^{2} \) |
| 37 | \( 1 + 2.37T + 37T^{2} \) |
| 41 | \( 1 - 3.38T + 41T^{2} \) |
| 43 | \( 1 - 2.22T + 43T^{2} \) |
| 47 | \( 1 + 0.738T + 47T^{2} \) |
| 53 | \( 1 + 9.97T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 + 4.21T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 0.960T + 71T^{2} \) |
| 73 | \( 1 + 6.48T + 73T^{2} \) |
| 79 | \( 1 + 0.640T + 79T^{2} \) |
| 83 | \( 1 + 2.87T + 83T^{2} \) |
| 89 | \( 1 - 6.58T + 89T^{2} \) |
| 97 | \( 1 - 5.45T + 97T^{2} \) |
| 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
−7.931641524573015900140183998407, −7.40815959670458495191768989731, −6.53996782735951982377975076951, −5.80792604369033429274710828350, −4.58266463795082213994023181478, −4.39542325000822825249679172639, −3.39443172542024476345284429325, −3.10533354686096728690681070841, −1.50511882055504459001970242317, −0.64354511035116624729505028066,
0.64354511035116624729505028066, 1.50511882055504459001970242317, 3.10533354686096728690681070841, 3.39443172542024476345284429325, 4.39542325000822825249679172639, 4.58266463795082213994023181478, 5.80792604369033429274710828350, 6.53996782735951982377975076951, 7.40815959670458495191768989731, 7.931641524573015900140183998407
