| L(s) = 1 | + 2.41·2-s − 2.82·3-s + 3.82·4-s − 6.82·6-s + 2.41·7-s + 4.41·8-s + 5.00·9-s + 6.41·11-s − 10.8·12-s + 13-s + 5.82·14-s + 2.99·16-s − 3.82·17-s + 12.0·18-s − 3.65·19-s − 6.82·21-s + 15.4·22-s − 3.17·23-s − 12.4·24-s + 2.41·26-s − 5.65·27-s + 9.24·28-s − 0.171·29-s + 1.24·31-s − 1.58·32-s − 18.1·33-s − 9.24·34-s + ⋯ |
| L(s) = 1 | + 1.70·2-s − 1.63·3-s + 1.91·4-s − 2.78·6-s + 0.912·7-s + 1.56·8-s + 1.66·9-s + 1.93·11-s − 3.12·12-s + 0.277·13-s + 1.55·14-s + 0.749·16-s − 0.928·17-s + 2.84·18-s − 0.838·19-s − 1.49·21-s + 3.30·22-s − 0.661·23-s − 2.54·24-s + 0.473·26-s − 1.08·27-s + 1.74·28-s − 0.0318·29-s + 0.223·31-s − 0.280·32-s − 3.15·33-s − 1.58·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.328797356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.328797356\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 3 | \( 1 + 2.82T + 3T^{2} \) |
| 7 | \( 1 - 2.41T + 7T^{2} \) |
| 11 | \( 1 - 6.41T + 11T^{2} \) |
| 17 | \( 1 + 3.82T + 17T^{2} \) |
| 19 | \( 1 + 3.65T + 19T^{2} \) |
| 23 | \( 1 + 3.17T + 23T^{2} \) |
| 29 | \( 1 + 0.171T + 29T^{2} \) |
| 31 | \( 1 - 1.24T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 - 0.828T + 43T^{2} \) |
| 47 | \( 1 - 3.58T + 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 + 2.75T + 67T^{2} \) |
| 71 | \( 1 + 9.31T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 6.89T + 83T^{2} \) |
| 89 | \( 1 + 8.48T + 89T^{2} \) |
| 97 | \( 1 + 0.828T + 97T^{2} \) |
| 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
−11.69118453302313807260667866569, −11.31619403154329265275011337116, −10.39365476407442346724634841233, −8.751752179507925201596945156451, −6.95476289889479535444152902723, −6.41937001867003862882005633423, −5.57823781576911185338576533511, −4.55722141207635688699446633808, −3.97464248680390603921789220431, −1.71771130309606177029769269375,
1.71771130309606177029769269375, 3.97464248680390603921789220431, 4.55722141207635688699446633808, 5.57823781576911185338576533511, 6.41937001867003862882005633423, 6.95476289889479535444152902723, 8.751752179507925201596945156451, 10.39365476407442346724634841233, 11.31619403154329265275011337116, 11.69118453302313807260667866569
