| L(s) = 1 | − 5.18·2-s + 3·3-s + 18.9·4-s − 15.5·6-s − 7·7-s − 56.5·8-s + 9·9-s − 35.9·11-s + 56.7·12-s − 45.2·13-s + 36.3·14-s + 142.·16-s − 113.·17-s − 46.6·18-s + 61.5·19-s − 21·21-s + 186.·22-s + 30.6·23-s − 169.·24-s + 234.·26-s + 27·27-s − 132.·28-s + 214.·29-s + 164.·31-s − 284.·32-s − 107.·33-s + 586.·34-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 0.577·3-s + 2.36·4-s − 1.05·6-s − 0.377·7-s − 2.49·8-s + 0.333·9-s − 0.985·11-s + 1.36·12-s − 0.965·13-s + 0.693·14-s + 2.21·16-s − 1.61·17-s − 0.611·18-s + 0.743·19-s − 0.218·21-s + 1.80·22-s + 0.277·23-s − 1.44·24-s + 1.77·26-s + 0.192·27-s − 0.892·28-s + 1.37·29-s + 0.951·31-s − 1.57·32-s − 0.569·33-s + 2.96·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6994781348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6994781348\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 5.18T + 8T^{2} \) |
| 11 | \( 1 + 35.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 45.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 113.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 61.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 30.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 214.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 164.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 410.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 309.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 29.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 483.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 295.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 416.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 151.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 89.5T + 3.00e5T^{2} \) |
| 71 | \( 1 - 714.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 323.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 297.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 90.2T + 7.04e5T^{2} \) |
| 97 | \( 1 - 492.T + 9.12e5T^{2} \) |
| 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
−10.01190512066983368266809708331, −9.666989415877792648823356457937, −8.630494883271673802102588273221, −8.045185100071386377380653081687, −7.14580161694160577542427093807, −6.42835531540617066271668782613, −4.80334313568686275141187759851, −2.91117834866568562006596571470, −2.22206766900301792340556247315, −0.62657069976590457054472821218,
0.62657069976590457054472821218, 2.22206766900301792340556247315, 2.91117834866568562006596571470, 4.80334313568686275141187759851, 6.42835531540617066271668782613, 7.14580161694160577542427093807, 8.045185100071386377380653081687, 8.630494883271673802102588273221, 9.666989415877792648823356457937, 10.01190512066983368266809708331
