| L(s) = 1 | + 2.41·2-s + 3.82·4-s + 5-s + 4.41·8-s + 2.41·10-s + 0.414·11-s + 2.24·13-s + 2.99·16-s − 4·17-s + 19-s + 3.82·20-s + 0.999·22-s + 5.58·23-s − 4·25-s + 5.41·26-s + 6.58·29-s + 6.24·31-s − 1.58·32-s − 9.65·34-s + 9.07·37-s + 2.41·38-s + 4.41·40-s − 3.17·41-s + 8.07·43-s + 1.58·44-s + 13.4·46-s − 4.41·47-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.91·4-s + 0.447·5-s + 1.56·8-s + 0.763·10-s + 0.124·11-s + 0.621·13-s + 0.749·16-s − 0.970·17-s + 0.229·19-s + 0.856·20-s + 0.213·22-s + 1.16·23-s − 0.800·25-s + 1.06·26-s + 1.22·29-s + 1.12·31-s − 0.280·32-s − 1.65·34-s + 1.49·37-s + 0.391·38-s + 0.697·40-s − 0.495·41-s + 1.23·43-s + 0.239·44-s + 1.98·46-s − 0.643·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.158163147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.158163147\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 - 0.414T + 11T^{2} \) |
| 13 | \( 1 - 2.24T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 23 | \( 1 - 5.58T + 23T^{2} \) |
| 29 | \( 1 - 6.58T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 - 9.07T + 37T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 - 8.07T + 43T^{2} \) |
| 47 | \( 1 + 4.41T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 - 6.82T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 1.17T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 - 1.41T + 79T^{2} \) |
| 83 | \( 1 - 9.24T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 3.17T + 97T^{2} \) |
| 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
−7.49500824789736660604985527017, −6.73040059535711733973809028427, −6.22993282204430689929273626205, −5.74250233009827087768408143642, −4.80593901626654231046243005465, −4.45419551497814991871024807533, −3.59471129435026268497584483393, −2.81435699176931005484273141129, −2.19846638818811275826015644718, −1.05568521557433465559119258008,
1.05568521557433465559119258008, 2.19846638818811275826015644718, 2.81435699176931005484273141129, 3.59471129435026268497584483393, 4.45419551497814991871024807533, 4.80593901626654231046243005465, 5.74250233009827087768408143642, 6.22993282204430689929273626205, 6.73040059535711733973809028427, 7.49500824789736660604985527017
