| L(s) = 1 | − i·2-s − 4-s − i·5-s − 3i·7-s + i·8-s − 3·9-s − 10-s − 3i·11-s − 3·14-s + 16-s + 4·17-s + 3i·18-s − 7i·19-s + i·20-s − 3·22-s + 4·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.447i·5-s − 1.13i·7-s + 0.353i·8-s − 9-s − 0.316·10-s − 0.904i·11-s − 0.801·14-s + 0.250·16-s + 0.970·17-s + 0.707i·18-s − 1.60i·19-s + 0.223i·20-s − 0.639·22-s + 0.834·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8443532246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8443532246\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 7iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 - 10iT - 31T^{2} \) |
| 37 | \( 1 - 3iT - 37T^{2} \) |
| 41 | \( 1 - 2iT - 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + iT - 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - iT - 89T^{2} \) |
| 97 | \( 1 + 16iT - 97T^{2} \) |
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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
−8.974472149526073955237103187319, −8.270018430517927848878049753117, −7.41553694019543771217817237170, −6.43754165048149942360579306641, −5.32998179087867168051080862613, −4.75028145790022338725931182794, −3.47909647332721996099127149099, −3.00934627706136889811015877011, −1.38069434395769789553772910976, −0.32955312861302782821520521777,
1.89730072661833625261073369982, 3.00216269545351557044257099271, 3.98759213212126458419970751422, 5.32716010435631143558311552002, 5.72066477715818156427266840820, 6.47484245721882194502602386002, 7.66805989713918990186968298301, 7.961908680755844214870090117511, 9.094911053156208925787671110171, 9.510048673850909021617120697475
