| L(s) = 1 | − 0.805·2-s − 1.95·3-s − 1.35·4-s + 1.57·6-s + 1.03·7-s + 2.69·8-s + 0.835·9-s + 1.80·11-s + 2.64·12-s + 2.36·13-s − 0.835·14-s + 0.529·16-s − 6.49·17-s − 0.672·18-s − 2.03·21-s − 1.45·22-s + 7.26·23-s − 5.28·24-s − 1.90·26-s + 4.23·27-s − 1.40·28-s − 2.22·29-s + 5.25·31-s − 5.82·32-s − 3.53·33-s + 5.23·34-s − 1.12·36-s + ⋯ |
| L(s) = 1 | − 0.569·2-s − 1.13·3-s − 0.675·4-s + 0.643·6-s + 0.391·7-s + 0.954·8-s + 0.278·9-s + 0.544·11-s + 0.764·12-s + 0.655·13-s − 0.223·14-s + 0.132·16-s − 1.57·17-s − 0.158·18-s − 0.443·21-s − 0.309·22-s + 1.51·23-s − 1.07·24-s − 0.373·26-s + 0.815·27-s − 0.264·28-s − 0.412·29-s + 0.943·31-s − 1.02·32-s − 0.615·33-s + 0.897·34-s − 0.188·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.805T + 2T^{2} \) |
| 3 | \( 1 + 1.95T + 3T^{2} \) |
| 7 | \( 1 - 1.03T + 7T^{2} \) |
| 11 | \( 1 - 1.80T + 11T^{2} \) |
| 13 | \( 1 - 2.36T + 13T^{2} \) |
| 17 | \( 1 + 6.49T + 17T^{2} \) |
| 23 | \( 1 - 7.26T + 23T^{2} \) |
| 29 | \( 1 + 2.22T + 29T^{2} \) |
| 31 | \( 1 - 5.25T + 31T^{2} \) |
| 37 | \( 1 + 6.67T + 37T^{2} \) |
| 41 | \( 1 + 4.43T + 41T^{2} \) |
| 43 | \( 1 + 6.26T + 43T^{2} \) |
| 47 | \( 1 - 8.38T + 47T^{2} \) |
| 53 | \( 1 + 0.0601T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 - 6.49T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 8.23T + 73T^{2} \) |
| 79 | \( 1 - 6.11T + 79T^{2} \) |
| 83 | \( 1 + 4.95T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
| 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
−7.24239962700986039330731540825, −6.75980836184153217254520379641, −6.05147350430980693557879131623, −5.23002290951543703674519816964, −4.73089331665848265441124875723, −4.11973252612361879986352532481, −3.10711085518798519290249605935, −1.77361538753419354148013067369, −0.965067941120881258736942863448, 0,
0.965067941120881258736942863448, 1.77361538753419354148013067369, 3.10711085518798519290249605935, 4.11973252612361879986352532481, 4.73089331665848265441124875723, 5.23002290951543703674519816964, 6.05147350430980693557879131623, 6.75980836184153217254520379641, 7.24239962700986039330731540825
