| L(s) = 1 | − 5-s + 4.24i·7-s + 1.41i·11-s − 4.24i·13-s + 2.82i·17-s + 4·19-s − 6·23-s + 25-s − 6·29-s + 8.48i·31-s − 4.24i·35-s + 4.24i·37-s − 9.89i·41-s − 8·43-s − 10.9·49-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.60i·7-s + 0.426i·11-s − 1.17i·13-s + 0.685i·17-s + 0.917·19-s − 1.25·23-s + 0.200·25-s − 1.11·29-s + 1.52i·31-s − 0.717i·35-s + 0.697i·37-s − 1.54i·41-s − 1.21·43-s − 1.57·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8128250899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8128250899\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 - 4.24iT - 7T^{2} \) |
| 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 4.24iT - 13T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 8.48iT - 31T^{2} \) |
| 37 | \( 1 - 4.24iT - 37T^{2} \) |
| 41 | \( 1 + 9.89iT - 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 - 1.41iT - 59T^{2} \) |
| 61 | \( 1 - 8.48iT - 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 - 8.48iT - 79T^{2} \) |
| 83 | \( 1 + 2.82iT - 83T^{2} \) |
| 89 | \( 1 - 7.07iT - 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.814664858414589688705655542096, −8.960821570036293786915612412004, −8.259802417940044400654304209173, −7.63099950730960849041949881857, −6.51102907695182650036893253737, −5.58668786992475249138219187811, −5.09465913109846719734902696805, −3.72389960287801075244983430054, −2.85963911508421866259496146784, −1.71515048087369980231577914394,
0.32018988291142652546488151022, 1.70792831392382837476634408737, 3.29978482382380789423498193042, 4.05845767456516893115500909095, 4.76573620317319398993794520093, 6.03956736963881357444584834575, 6.91303808038793915420660093173, 7.60089067911024098961765696432, 8.164200710749238443093188593129, 9.545668919304885953895126202993
