| L(s) = 1 | + 1.56i·2-s − 0.438·4-s + i·7-s + 2.43i·8-s + 11-s + 4.56i·13-s − 1.56·14-s − 4.68·16-s − 5.56i·17-s − 3·19-s + 1.56i·22-s + 2.43i·23-s − 7.12·26-s − 0.438i·28-s − 3.12·29-s + ⋯ |
| L(s) = 1 | + 1.10i·2-s − 0.219·4-s + 0.377i·7-s + 0.862i·8-s + 0.301·11-s + 1.26i·13-s − 0.417·14-s − 1.17·16-s − 1.34i·17-s − 0.688·19-s + 0.332i·22-s + 0.508i·23-s − 1.39·26-s − 0.0828i·28-s − 0.579·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.244125541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.244125541\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.56iT - 2T^{2} \) |
| 7 | \( 1 - iT - 7T^{2} \) |
| 13 | \( 1 - 4.56iT - 13T^{2} \) |
| 17 | \( 1 + 5.56iT - 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 - 2.43iT - 23T^{2} \) |
| 29 | \( 1 + 3.12T + 29T^{2} \) |
| 31 | \( 1 + 7.68T + 31T^{2} \) |
| 37 | \( 1 - 9.80iT - 37T^{2} \) |
| 41 | \( 1 + 4.68T + 41T^{2} \) |
| 43 | \( 1 - 7.68iT - 43T^{2} \) |
| 47 | \( 1 + 9.56iT - 47T^{2} \) |
| 53 | \( 1 - 7.12iT - 53T^{2} \) |
| 59 | \( 1 - 6.43T + 59T^{2} \) |
| 61 | \( 1 + 3.43T + 61T^{2} \) |
| 67 | \( 1 - 5.68iT - 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 0.246iT - 73T^{2} \) |
| 79 | \( 1 - 3.31T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 8.87T + 89T^{2} \) |
| 97 | \( 1 + 6.12iT - 97T^{2} \) |
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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
−9.112720848933867194916486898527, −8.574669591830102449243666552023, −7.65586307660219929821938463274, −6.94529059754499397057324590947, −6.50842325991172751989809607298, −5.54858663440858988821234396419, −4.92301312338470521036968653056, −3.94909057145897918224884789337, −2.67491916111918530759998873861, −1.70972079889137685224160578752,
0.38540848791446555181716825898, 1.63450027625765442858709780723, 2.48138320501381293773530621399, 3.68789924791496370758637845471, 3.94128463182062025940254542045, 5.29046983254217008529237472935, 6.13870378759931560984756704690, 6.98224722697097218630277832560, 7.78808449218648055462591957042, 8.644559402712053246348349328356
