| L(s) = 1 | + 3-s + 4.35·5-s − 4.92·7-s + 9-s + 1.24·11-s + 5.88·13-s + 4.35·15-s − 2.94·17-s − 3.68·19-s − 4.92·21-s − 0.211·23-s + 13.9·25-s + 27-s + 5.01·29-s + 2.92·31-s + 1.24·33-s − 21.4·35-s − 4.27·37-s + 5.88·39-s − 2.80·41-s − 1.34·43-s + 4.35·45-s + 8.33·47-s + 17.2·49-s − 2.94·51-s + 5.50·53-s + 5.44·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.94·5-s − 1.86·7-s + 0.333·9-s + 0.376·11-s + 1.63·13-s + 1.12·15-s − 0.713·17-s − 0.844·19-s − 1.07·21-s − 0.0440·23-s + 2.79·25-s + 0.192·27-s + 0.931·29-s + 0.525·31-s + 0.217·33-s − 3.62·35-s − 0.702·37-s + 0.942·39-s − 0.438·41-s − 0.204·43-s + 0.649·45-s + 1.21·47-s + 2.45·49-s − 0.411·51-s + 0.756·53-s + 0.734·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.326545920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.326545920\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 251 | \( 1 + T \) |
| good | 5 | \( 1 - 4.35T + 5T^{2} \) |
| 7 | \( 1 + 4.92T + 7T^{2} \) |
| 11 | \( 1 - 1.24T + 11T^{2} \) |
| 13 | \( 1 - 5.88T + 13T^{2} \) |
| 17 | \( 1 + 2.94T + 17T^{2} \) |
| 19 | \( 1 + 3.68T + 19T^{2} \) |
| 23 | \( 1 + 0.211T + 23T^{2} \) |
| 29 | \( 1 - 5.01T + 29T^{2} \) |
| 31 | \( 1 - 2.92T + 31T^{2} \) |
| 37 | \( 1 + 4.27T + 37T^{2} \) |
| 41 | \( 1 + 2.80T + 41T^{2} \) |
| 43 | \( 1 + 1.34T + 43T^{2} \) |
| 47 | \( 1 - 8.33T + 47T^{2} \) |
| 53 | \( 1 - 5.50T + 53T^{2} \) |
| 59 | \( 1 - 9.06T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 4.09T + 67T^{2} \) |
| 71 | \( 1 - 3.07T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 - 8.87T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 2.62T + 97T^{2} \) |
| 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
−8.529058404343591900757396415250, −6.96678251741759537451999639227, −6.52844198309224045466477121601, −6.16646378011730082214074312908, −5.49115785917007196080979174133, −4.28966836694869953959858215902, −3.46228448031918670045240410252, −2.72590162699598484891035635515, −2.00837421050463432297048756749, −0.956844682739404990843189778141,
0.956844682739404990843189778141, 2.00837421050463432297048756749, 2.72590162699598484891035635515, 3.46228448031918670045240410252, 4.28966836694869953959858215902, 5.49115785917007196080979174133, 6.16646378011730082214074312908, 6.52844198309224045466477121601, 6.96678251741759537451999639227, 8.529058404343591900757396415250
