| L(s) = 1 | + 0.356·3-s − 2.46·7-s − 2.87·9-s + 1.61·11-s + 2.62·13-s + 2.58·17-s + 4.02·19-s − 0.878·21-s − 23-s − 2.09·27-s − 7.08·29-s − 4.58·31-s + 0.575·33-s − 2.96·37-s + 0.935·39-s − 5.71·41-s + 2.30·43-s − 6.88·47-s − 0.929·49-s + 0.922·51-s + 6.76·53-s + 1.43·57-s + 2.53·59-s + 9.25·61-s + 7.07·63-s + 15.7·67-s − 0.356·69-s + ⋯ |
| L(s) = 1 | + 0.205·3-s − 0.931·7-s − 0.957·9-s + 0.487·11-s + 0.728·13-s + 0.627·17-s + 0.923·19-s − 0.191·21-s − 0.208·23-s − 0.402·27-s − 1.31·29-s − 0.823·31-s + 0.100·33-s − 0.486·37-s + 0.149·39-s − 0.891·41-s + 0.351·43-s − 1.00·47-s − 0.132·49-s + 0.129·51-s + 0.929·53-s + 0.190·57-s + 0.329·59-s + 1.18·61-s + 0.891·63-s + 1.92·67-s − 0.0429·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 0.356T + 3T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 - 1.61T + 11T^{2} \) |
| 13 | \( 1 - 2.62T + 13T^{2} \) |
| 17 | \( 1 - 2.58T + 17T^{2} \) |
| 19 | \( 1 - 4.02T + 19T^{2} \) |
| 29 | \( 1 + 7.08T + 29T^{2} \) |
| 31 | \( 1 + 4.58T + 31T^{2} \) |
| 37 | \( 1 + 2.96T + 37T^{2} \) |
| 41 | \( 1 + 5.71T + 41T^{2} \) |
| 43 | \( 1 - 2.30T + 43T^{2} \) |
| 47 | \( 1 + 6.88T + 47T^{2} \) |
| 53 | \( 1 - 6.76T + 53T^{2} \) |
| 59 | \( 1 - 2.53T + 59T^{2} \) |
| 61 | \( 1 - 9.25T + 61T^{2} \) |
| 67 | \( 1 - 15.7T + 67T^{2} \) |
| 71 | \( 1 + 5.25T + 71T^{2} \) |
| 73 | \( 1 + 6.03T + 73T^{2} \) |
| 79 | \( 1 + 1.35T + 79T^{2} \) |
| 83 | \( 1 - 8.59T + 83T^{2} \) |
| 89 | \( 1 + 8.84T + 89T^{2} \) |
| 97 | \( 1 + 8.01T + 97T^{2} \) |
| 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
−8.069752169682404660577578234319, −7.16811823927347339548006358213, −6.53090523905569060747220303009, −5.68109020117942368697194335742, −5.27882286286787001564779537593, −3.75071408261960247271556006558, −3.53639025723533032283837588392, −2.56013718914314383544108047239, −1.36886688188199271753749348557, 0,
1.36886688188199271753749348557, 2.56013718914314383544108047239, 3.53639025723533032283837588392, 3.75071408261960247271556006558, 5.27882286286787001564779537593, 5.68109020117942368697194335742, 6.53090523905569060747220303009, 7.16811823927347339548006358213, 8.069752169682404660577578234319
