| L(s) = 1 | − 2.81·2-s + 3-s + 5.92·4-s + 5-s − 2.81·6-s − 7-s − 11.0·8-s + 9-s − 2.81·10-s + 1.05·11-s + 5.92·12-s − 4.56·13-s + 2.81·14-s + 15-s + 19.2·16-s − 7.71·17-s − 2.81·18-s + 4.87·19-s + 5.92·20-s − 21-s − 2.95·22-s + 23-s − 11.0·24-s + 25-s + 12.8·26-s + 27-s − 5.92·28-s + ⋯ |
| L(s) = 1 | − 1.99·2-s + 0.577·3-s + 2.96·4-s + 0.447·5-s − 1.14·6-s − 0.377·7-s − 3.91·8-s + 0.333·9-s − 0.890·10-s + 0.316·11-s + 1.71·12-s − 1.26·13-s + 0.752·14-s + 0.258·15-s + 4.82·16-s − 1.87·17-s − 0.663·18-s + 1.11·19-s + 1.32·20-s − 0.218·21-s − 0.630·22-s + 0.208·23-s − 2.25·24-s + 0.200·25-s + 2.52·26-s + 0.192·27-s − 1.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8062268038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8062268038\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.81T + 2T^{2} \) |
| 11 | \( 1 - 1.05T + 11T^{2} \) |
| 13 | \( 1 + 4.56T + 13T^{2} \) |
| 17 | \( 1 + 7.71T + 17T^{2} \) |
| 19 | \( 1 - 4.87T + 19T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 + 1.49T + 31T^{2} \) |
| 37 | \( 1 + 3.18T + 37T^{2} \) |
| 41 | \( 1 + 2.38T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 - 9.35T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 - 6.28T + 59T^{2} \) |
| 61 | \( 1 - 4.04T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 9.36T + 73T^{2} \) |
| 79 | \( 1 - 7.16T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + 2.23T + 89T^{2} \) |
| 97 | \( 1 - 2.02T + 97T^{2} \) |
| 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
−9.180583366385093428145826149984, −8.455793295871984035889715934557, −7.35220532443313122030508106305, −7.17600551189034182686163126800, −6.31047781996980640499350815014, −5.30976085692280031101063467421, −3.70741795157735158237013108073, −2.48680709709111121075840192098, −2.14511880946285046050611689834, −0.72270746181007452193646789728,
0.72270746181007452193646789728, 2.14511880946285046050611689834, 2.48680709709111121075840192098, 3.70741795157735158237013108073, 5.30976085692280031101063467421, 6.31047781996980640499350815014, 7.17600551189034182686163126800, 7.35220532443313122030508106305, 8.455793295871984035889715934557, 9.180583366385093428145826149984
