| L(s) = 1 | + i·3-s − 0.547i·7-s − 9-s − 1.33·11-s + 0.302i·13-s − 4.04i·17-s + 5.63·19-s + 0.547·21-s + 0.245i·23-s − i·27-s + 1.36·29-s − 3.53·31-s − 1.33i·33-s + 2.18i·37-s − 0.302·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.206i·7-s − 0.333·9-s − 0.403·11-s + 0.0837i·13-s − 0.980i·17-s + 1.29·19-s + 0.119·21-s + 0.0511i·23-s − 0.192i·27-s + 0.254·29-s − 0.635·31-s − 0.232i·33-s + 0.359i·37-s − 0.0483·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.029478197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.029478197\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 0.547iT - 7T^{2} \) |
| 11 | \( 1 + 1.33T + 11T^{2} \) |
| 13 | \( 1 - 0.302iT - 13T^{2} \) |
| 17 | \( 1 + 4.04iT - 17T^{2} \) |
| 19 | \( 1 - 5.63T + 19T^{2} \) |
| 23 | \( 1 - 0.245iT - 23T^{2} \) |
| 29 | \( 1 - 1.36T + 29T^{2} \) |
| 31 | \( 1 + 3.53T + 31T^{2} \) |
| 37 | \( 1 - 2.18iT - 37T^{2} \) |
| 41 | \( 1 + 9.80T + 41T^{2} \) |
| 43 | \( 1 + 8.35iT - 43T^{2} \) |
| 47 | \( 1 - 10.4iT - 47T^{2} \) |
| 53 | \( 1 + 7.69iT - 53T^{2} \) |
| 59 | \( 1 + 4.15T + 59T^{2} \) |
| 61 | \( 1 - 2.07T + 61T^{2} \) |
| 67 | \( 1 + 8.48iT - 67T^{2} \) |
| 71 | \( 1 + 9.18T + 71T^{2} \) |
| 73 | \( 1 - 10.9iT - 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 6.65iT - 83T^{2} \) |
| 89 | \( 1 + 4.97T + 89T^{2} \) |
| 97 | \( 1 + 4.06iT - 97T^{2} \) |
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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
−7.55342747355424183777895425876, −7.21428011448621425014554074842, −6.28203357336717077751132596558, −5.38072304162661199944322365494, −5.01739553207832140550772717918, −4.13923933637649239981661807117, −3.30398393659682250450839338181, −2.69903437106796502491675153233, −1.50254873826807905607202347679, −0.26029167982790399538179866666,
1.09282416721942664408284934566, 1.95511334041563168843364298943, 2.89603754961245393033766123751, 3.59529904898894483961837118580, 4.57747580275926853873502843107, 5.46362719914783745333269887872, 5.88005632833217719334623035757, 6.77688483478967115285529115069, 7.37316794445824221505994836775, 8.004705230797277307963708453095
