| L(s) = 1 | − i·2-s − 4-s − 3.11·5-s + (−0.672 + 2.55i)7-s + i·8-s + 3.11i·10-s + 2.40i·11-s + 5.44i·13-s + (2.55 + 0.672i)14-s + 16-s − 1.29·17-s − i·19-s + 3.11·20-s + 2.40·22-s − 4.76i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 1.39·5-s + (−0.254 + 0.967i)7-s + 0.353i·8-s + 0.984i·10-s + 0.724i·11-s + 1.51i·13-s + (0.683 + 0.179i)14-s + 0.250·16-s − 0.315·17-s − 0.229i·19-s + 0.696·20-s + 0.512·22-s − 0.993i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1290777409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1290777409\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.672 - 2.55i)T \) |
| 19 | \( 1 + iT \) |
| good | 5 | \( 1 + 3.11T + 5T^{2} \) |
| 11 | \( 1 - 2.40iT - 11T^{2} \) |
| 13 | \( 1 - 5.44iT - 13T^{2} \) |
| 17 | \( 1 + 1.29T + 17T^{2} \) |
| 23 | \( 1 + 4.76iT - 23T^{2} \) |
| 29 | \( 1 + 0.688iT - 29T^{2} \) |
| 31 | \( 1 - 1.87iT - 31T^{2} \) |
| 37 | \( 1 + 3.32T + 37T^{2} \) |
| 41 | \( 1 - 0.802T + 41T^{2} \) |
| 43 | \( 1 - 1.43T + 43T^{2} \) |
| 47 | \( 1 + 3.20T + 47T^{2} \) |
| 53 | \( 1 + 12.4iT - 53T^{2} \) |
| 59 | \( 1 + 1.74T + 59T^{2} \) |
| 61 | \( 1 + 0.829iT - 61T^{2} \) |
| 67 | \( 1 + 0.826T + 67T^{2} \) |
| 71 | \( 1 - 5.35iT - 71T^{2} \) |
| 73 | \( 1 - 0.532iT - 73T^{2} \) |
| 79 | \( 1 - 0.338T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 15.1iT - 97T^{2} \) |
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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
−8.700611083338094917306490153067, −8.093562865478207843055261778965, −7.05214002453618125861522608602, −6.47880198216107314264607102649, −5.14592517761550456221726077696, −4.42504518075860448044190273841, −3.77698830453919129522030348181, −2.71275373586388174685959824909, −1.80830806538377308638770630910, −0.05571472488471992934492569487,
0.952306422471875036763471979902, 3.12641517691504189308111342568, 3.67353154711026256899855384540, 4.47511106016637285142823082656, 5.44661307488346432466024126461, 6.27102714446656919993998775122, 7.27439182317319517235542766034, 7.70942143377328055405428029983, 8.222166536437721999577251238988, 9.074906330564056195375035644183
