| L(s) = 1 | − 2-s + 2.29i·3-s + 4-s + (1.91 + 1.14i)5-s − 2.29i·6-s − 1.25·7-s − 8-s − 2.25·9-s + (−1.91 − 1.14i)10-s + 2i·11-s + 2.29i·12-s + 1.25·14-s + (−2.62 + 4.40i)15-s + 16-s + 4.80i·17-s + 2.25·18-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.32i·3-s + 0.5·4-s + (0.858 + 0.512i)5-s − 0.935i·6-s − 0.474·7-s − 0.353·8-s − 0.751·9-s + (−0.607 − 0.362i)10-s + 0.603i·11-s + 0.661i·12-s + 0.335·14-s + (−0.678 + 1.13i)15-s + 0.250·16-s + 1.16i·17-s + 0.531·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0497i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9867739878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9867739878\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + (-1.91 - 1.14i)T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.29iT - 3T^{2} \) |
| 7 | \( 1 + 1.25T + 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 17 | \( 1 - 4.80iT - 17T^{2} \) |
| 19 | \( 1 + 5.09iT - 19T^{2} \) |
| 23 | \( 1 - 2.58iT - 23T^{2} \) |
| 29 | \( 1 + 5.09T + 29T^{2} \) |
| 31 | \( 1 + 8.58iT - 31T^{2} \) |
| 37 | \( 1 + 7.83T + 37T^{2} \) |
| 41 | \( 1 - 9.67iT - 41T^{2} \) |
| 43 | \( 1 - 10.8iT - 43T^{2} \) |
| 47 | \( 1 + 2.74T + 47T^{2} \) |
| 53 | \( 1 + 2.58iT - 53T^{2} \) |
| 59 | \( 1 - 5.09iT - 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 - 8.58T + 67T^{2} \) |
| 71 | \( 1 + 5.38iT - 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 5.09iT - 89T^{2} \) |
| 97 | \( 1 + 6.26T + 97T^{2} \) |
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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
−9.822431181045793824473568196446, −9.323100345922509259218784223019, −8.448485857019121896037091017882, −7.36033841556919623788095249464, −6.53898808139919509952003751378, −5.76143694277725784126230476505, −4.82104751518197306901767156390, −3.78309185369648996509562472961, −2.87519998283783997080156397423, −1.74137854125933420682836548406,
0.45747550380330726957066580989, 1.55053457741559147773084123900, 2.35701403968438570925594314934, 3.53823236805570152209569160831, 5.24796558956238118942830053758, 5.86837662351405034936501209409, 6.85026498663319081715947861420, 7.15158003207073849179233258512, 8.369481804636732488528884591229, 8.695279702773658386071463506655
