| L(s) = 1 | − 1.59·3-s + 4.51·7-s − 0.460·9-s − 0.593·11-s + 4.05·13-s + 5.32·17-s − 19-s − 7.19·21-s − 7.46·23-s + 5.51·27-s − 2.32·29-s − 6.97·31-s + 0.945·33-s − 3.40·37-s − 6.46·39-s − 7.38·41-s − 6.18·43-s − 12.7·47-s + 13.3·49-s − 8.48·51-s − 8.84·53-s + 1.59·57-s − 4.10·59-s + 9.89·61-s − 2.07·63-s − 12.0·67-s + 11.8·69-s + ⋯ |
| L(s) = 1 | − 0.920·3-s + 1.70·7-s − 0.153·9-s − 0.178·11-s + 1.12·13-s + 1.29·17-s − 0.229·19-s − 1.56·21-s − 1.55·23-s + 1.06·27-s − 0.432·29-s − 1.25·31-s + 0.164·33-s − 0.560·37-s − 1.03·39-s − 1.15·41-s − 0.943·43-s − 1.85·47-s + 1.91·49-s − 1.18·51-s − 1.21·53-s + 0.211·57-s − 0.534·59-s + 1.26·61-s − 0.261·63-s − 1.47·67-s + 1.43·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.59T + 3T^{2} \) |
| 7 | \( 1 - 4.51T + 7T^{2} \) |
| 11 | \( 1 + 0.593T + 11T^{2} \) |
| 13 | \( 1 - 4.05T + 13T^{2} \) |
| 17 | \( 1 - 5.32T + 17T^{2} \) |
| 23 | \( 1 + 7.46T + 23T^{2} \) |
| 29 | \( 1 + 2.32T + 29T^{2} \) |
| 31 | \( 1 + 6.97T + 31T^{2} \) |
| 37 | \( 1 + 3.40T + 37T^{2} \) |
| 41 | \( 1 + 7.38T + 41T^{2} \) |
| 43 | \( 1 + 6.18T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 + 8.84T + 53T^{2} \) |
| 59 | \( 1 + 4.10T + 59T^{2} \) |
| 61 | \( 1 - 9.89T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 7.10T + 71T^{2} \) |
| 73 | \( 1 - 5.27T + 73T^{2} \) |
| 79 | \( 1 - 3.96T + 79T^{2} \) |
| 83 | \( 1 + 7.53T + 83T^{2} \) |
| 89 | \( 1 - 9.19T + 89T^{2} \) |
| 97 | \( 1 - 6.42T + 97T^{2} \) |
| 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
−7.71237860854122023376228853216, −6.69756419990994998124462742887, −6.00404948781736422108877412415, −5.32890786270312025157035507447, −5.01470698388344992791718562631, −4.00800165533448288324748627998, −3.27944135676774517536581845278, −1.86966530535767396221116226494, −1.36976245838831743602217666192, 0,
1.36976245838831743602217666192, 1.86966530535767396221116226494, 3.27944135676774517536581845278, 4.00800165533448288324748627998, 5.01470698388344992791718562631, 5.32890786270312025157035507447, 6.00404948781736422108877412415, 6.69756419990994998124462742887, 7.71237860854122023376228853216
