| L(s) = 1 | − 5-s + 2·7-s + 0.267·13-s − 3.73·17-s + 4·19-s + 3.46·23-s − 4·25-s − 5.73·29-s + 6.92·31-s − 2·35-s − 8.46·37-s − 4.26·41-s − 2·43-s + 2.92·47-s − 3·49-s − 5.92·53-s − 7.46·59-s + 2.92·61-s − 0.267·65-s − 7.46·67-s − 10.3·71-s + 1.07·73-s + 10.9·79-s + 13.8·83-s + 3.73·85-s + 11.3·89-s + 0.535·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s + 0.0743·13-s − 0.905·17-s + 0.917·19-s + 0.722·23-s − 0.800·25-s − 1.06·29-s + 1.24·31-s − 0.338·35-s − 1.39·37-s − 0.666·41-s − 0.304·43-s + 0.427·47-s − 0.428·49-s − 0.814·53-s − 0.971·59-s + 0.374·61-s − 0.0332·65-s − 0.911·67-s − 1.23·71-s + 0.125·73-s + 1.22·79-s + 1.52·83-s + 0.404·85-s + 1.20·89-s + 0.0561·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 13 | \( 1 - 0.267T + 13T^{2} \) |
| 17 | \( 1 + 3.73T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 5.73T + 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 + 8.46T + 37T^{2} \) |
| 41 | \( 1 + 4.26T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 2.92T + 47T^{2} \) |
| 53 | \( 1 + 5.92T + 53T^{2} \) |
| 59 | \( 1 + 7.46T + 59T^{2} \) |
| 61 | \( 1 - 2.92T + 61T^{2} \) |
| 67 | \( 1 + 7.46T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 1.07T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
| 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
−7.58601323434731437603655221127, −6.78554632976035981440262720094, −6.10027312308682762806398757333, −5.12999887146673790198330235900, −4.76185622136655974840799333121, −3.84126138510495712576197081127, −3.16595993584504138580785156029, −2.12095249064722386024521893512, −1.29718123175720822156992802372, 0,
1.29718123175720822156992802372, 2.12095249064722386024521893512, 3.16595993584504138580785156029, 3.84126138510495712576197081127, 4.76185622136655974840799333121, 5.12999887146673790198330235900, 6.10027312308682762806398757333, 6.78554632976035981440262720094, 7.58601323434731437603655221127
