| L(s) = 1 | − 0.874·3-s + 4.39·7-s − 2.23·9-s + 11-s − 5.45·13-s − 1.33·17-s + 6.79·19-s − 3.84·21-s − 2.24·23-s + 4.57·27-s − 5.19·29-s − 2.58·31-s − 0.874·33-s + 1.13·37-s + 4.77·39-s − 5.31·41-s + 4.22·43-s − 8.05·47-s + 12.2·49-s + 1.16·51-s + 13.1·53-s − 5.94·57-s + 2.95·59-s − 10.6·61-s − 9.81·63-s + 0.994·67-s + 1.96·69-s + ⋯ |
| L(s) = 1 | − 0.504·3-s + 1.66·7-s − 0.744·9-s + 0.301·11-s − 1.51·13-s − 0.324·17-s + 1.55·19-s − 0.838·21-s − 0.468·23-s + 0.881·27-s − 0.964·29-s − 0.464·31-s − 0.152·33-s + 0.186·37-s + 0.764·39-s − 0.829·41-s + 0.643·43-s − 1.17·47-s + 1.75·49-s + 0.163·51-s + 1.80·53-s − 0.786·57-s + 0.385·59-s − 1.35·61-s − 1.23·63-s + 0.121·67-s + 0.236·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 0.874T + 3T^{2} \) |
| 7 | \( 1 - 4.39T + 7T^{2} \) |
| 13 | \( 1 + 5.45T + 13T^{2} \) |
| 17 | \( 1 + 1.33T + 17T^{2} \) |
| 19 | \( 1 - 6.79T + 19T^{2} \) |
| 23 | \( 1 + 2.24T + 23T^{2} \) |
| 29 | \( 1 + 5.19T + 29T^{2} \) |
| 31 | \( 1 + 2.58T + 31T^{2} \) |
| 37 | \( 1 - 1.13T + 37T^{2} \) |
| 41 | \( 1 + 5.31T + 41T^{2} \) |
| 43 | \( 1 - 4.22T + 43T^{2} \) |
| 47 | \( 1 + 8.05T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 - 2.95T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 0.994T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 + 8.42T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 9.36T + 83T^{2} \) |
| 89 | \( 1 + 9.47T + 89T^{2} \) |
| 97 | \( 1 + 4.10T + 97T^{2} \) |
| 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
−7.40804292239927756658726337461, −6.90671234217209056825202839913, −5.71866526249771822325357424590, −5.36887217829664411263217590792, −4.80145361123876081683340000894, −4.05004323764517965687680515724, −2.94841629549116412442634721256, −2.12566991424100942240582616715, −1.25791152471458398011092572245, 0,
1.25791152471458398011092572245, 2.12566991424100942240582616715, 2.94841629549116412442634721256, 4.05004323764517965687680515724, 4.80145361123876081683340000894, 5.36887217829664411263217590792, 5.71866526249771822325357424590, 6.90671234217209056825202839913, 7.40804292239927756658726337461
