| L(s) = 1 | + i·2-s + 4-s + 2i·7-s + 3i·8-s − 11-s − 2i·13-s − 2·14-s − 16-s − 2i·17-s + 6·19-s − i·22-s + 4i·23-s + 2·26-s + 2i·28-s + 6·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.5·4-s + 0.755i·7-s + 1.06i·8-s − 0.301·11-s − 0.554i·13-s − 0.534·14-s − 0.250·16-s − 0.485i·17-s + 1.37·19-s − 0.213i·22-s + 0.834i·23-s + 0.392·26-s + 0.377i·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.113552536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.113552536\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT - 2T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−9.064772618816293519184799270513, −8.085054617155687207817875365553, −7.77650150645533024114670686994, −6.81458709377288916645525663819, −6.14786168815816536174015883674, −5.33300021623181321636655425015, −4.84623218618087504689168216774, −3.19536578112855500461539776202, −2.68241370116663028907022659859, −1.35300550056206041348954484465,
0.72821514763032145207897256027, 1.80067723740961108311090070147, 2.82947706574943125181324552895, 3.66254637291172556164165333417, 4.47725309699010482868661396691, 5.52324681384973635070138776531, 6.58400181546080387468159689340, 7.05144308678205909703116615342, 7.86460325274192278819449964194, 8.757033627962422923244267847932
