| L(s) = 1 | − i·2-s + 0.381i·3-s − 4-s + 0.381·6-s − 3i·7-s + i·8-s + 2.85·9-s + 4.23·11-s − 0.381i·12-s − i·13-s − 3·14-s + 16-s + 1.14i·17-s − 2.85i·18-s − 5.85·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.220i·3-s − 0.5·4-s + 0.155·6-s − 1.13i·7-s + 0.353i·8-s + 0.951·9-s + 1.27·11-s − 0.110i·12-s − 0.277i·13-s − 0.801·14-s + 0.250·16-s + 0.277i·17-s − 0.672i·18-s − 1.34·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.722037243\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.722037243\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 0.381iT - 3T^{2} \) |
| 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 1.14iT - 17T^{2} \) |
| 19 | \( 1 + 5.85T + 19T^{2} \) |
| 23 | \( 1 - 1.76iT - 23T^{2} \) |
| 29 | \( 1 - 9.47T + 29T^{2} \) |
| 31 | \( 1 + 0.236T + 31T^{2} \) |
| 37 | \( 1 + 8.32iT - 37T^{2} \) |
| 41 | \( 1 - 1.47T + 41T^{2} \) |
| 43 | \( 1 - 6.23iT - 43T^{2} \) |
| 47 | \( 1 + 11.9iT - 47T^{2} \) |
| 53 | \( 1 + 10.4iT - 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 + 8.85T + 61T^{2} \) |
| 67 | \( 1 + 10.2iT - 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 + 7.70iT - 73T^{2} \) |
| 79 | \( 1 - 7.23T + 79T^{2} \) |
| 83 | \( 1 - 4.52iT - 83T^{2} \) |
| 89 | \( 1 + 4.47T + 89T^{2} \) |
| 97 | \( 1 - 9.56iT - 97T^{2} \) |
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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
−9.684904124025644434922018586049, −8.880163212492177630404272659946, −7.961716926358658183648969566382, −6.96627497438097932200129665552, −6.31484923910832774232050583647, −4.84112942939226211244088273030, −4.10728643025840240001546436719, −3.55823356814939750772689012486, −1.95776957835675333042976725576, −0.859212574906169052191574286132,
1.34650779124563106976574123055, 2.66779935051917070893247625101, 4.15269854597057949304626604280, 4.73586548805721697813321347915, 6.06832231843345329201761513885, 6.48742570445656358083308216480, 7.29399573101097526938258364320, 8.426390028753851260792260410507, 8.873882486207604783752026943169, 9.684861544738837969581865009524
