| L(s) = 1 | + 3.07·3-s + 5-s − 2.07·7-s + 6.48·9-s + 5.07·11-s − 3.48·13-s + 3.07·15-s + 6.48·17-s − 7.48·19-s − 6.40·21-s + 23-s + 25-s + 10.7·27-s − 1.56·29-s + 0.0791·31-s + 15.6·33-s − 2.07·35-s − 9.71·37-s − 10.7·39-s − 0.480·41-s + 8·43-s + 6.48·45-s + 6.96·47-s − 2.67·49-s + 19.9·51-s + 11.7·53-s + 5.07·55-s + ⋯ |
| L(s) = 1 | + 1.77·3-s + 0.447·5-s − 0.785·7-s + 2.16·9-s + 1.53·11-s − 0.965·13-s + 0.795·15-s + 1.57·17-s − 1.71·19-s − 1.39·21-s + 0.208·23-s + 0.200·25-s + 2.06·27-s − 0.289·29-s + 0.0142·31-s + 2.72·33-s − 0.351·35-s − 1.59·37-s − 1.71·39-s − 0.0751·41-s + 1.21·43-s + 0.966·45-s + 1.01·47-s − 0.382·49-s + 2.79·51-s + 1.60·53-s + 0.684·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.034834952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.034834952\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 3.07T + 3T^{2} \) |
| 7 | \( 1 + 2.07T + 7T^{2} \) |
| 11 | \( 1 - 5.07T + 11T^{2} \) |
| 13 | \( 1 + 3.48T + 13T^{2} \) |
| 17 | \( 1 - 6.48T + 17T^{2} \) |
| 19 | \( 1 + 7.48T + 19T^{2} \) |
| 29 | \( 1 + 1.56T + 29T^{2} \) |
| 31 | \( 1 - 0.0791T + 31T^{2} \) |
| 37 | \( 1 + 9.71T + 37T^{2} \) |
| 41 | \( 1 + 0.480T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 6.96T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 7.88T + 61T^{2} \) |
| 67 | \( 1 + 9.71T + 67T^{2} \) |
| 71 | \( 1 + 9.67T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 4.59T + 83T^{2} \) |
| 89 | \( 1 - 8.31T + 89T^{2} \) |
| 97 | \( 1 - 7.23T + 97T^{2} \) |
| 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
−9.926146718215994961850415868461, −9.029566655532443377009804811765, −8.778106974065532144620875794308, −7.54378579353636160191221604105, −6.93154349237669741552996497672, −5.90556893088585129142762414397, −4.35912673720096826832977216483, −3.56742418908269208095011621392, −2.66584819461882898238006159893, −1.58709543811419562535628536652,
1.58709543811419562535628536652, 2.66584819461882898238006159893, 3.56742418908269208095011621392, 4.35912673720096826832977216483, 5.90556893088585129142762414397, 6.93154349237669741552996497672, 7.54378579353636160191221604105, 8.778106974065532144620875794308, 9.029566655532443377009804811765, 9.926146718215994961850415868461
