| L(s) = 1 | − 2-s + 4-s + (−1.93 − 1.80i)7-s − 8-s − 3.87i·11-s − 1.60·13-s + (1.93 + 1.80i)14-s + 16-s − 8.11i·17-s − 2.63i·19-s + 3.87i·22-s − 5.47·23-s + 1.60·26-s + (−1.93 − 1.80i)28-s + 5.47i·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + (−0.732 − 0.681i)7-s − 0.353·8-s − 1.16i·11-s − 0.445·13-s + (0.517 + 0.481i)14-s + 0.250·16-s − 1.96i·17-s − 0.605i·19-s + 0.825i·22-s − 1.14·23-s + 0.314·26-s + (−0.366 − 0.340i)28-s + 1.01i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 - 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.826 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2135315847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2135315847\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.93 + 1.80i)T \) |
| good | 11 | \( 1 + 3.87iT - 11T^{2} \) |
| 13 | \( 1 + 1.60T + 13T^{2} \) |
| 17 | \( 1 + 8.11iT - 17T^{2} \) |
| 19 | \( 1 + 2.63iT - 19T^{2} \) |
| 23 | \( 1 + 5.47T + 23T^{2} \) |
| 29 | \( 1 - 5.47iT - 29T^{2} \) |
| 31 | \( 1 + 3.73iT - 31T^{2} \) |
| 37 | \( 1 - 4.51iT - 37T^{2} \) |
| 41 | \( 1 - 1.60T + 41T^{2} \) |
| 43 | \( 1 - 10.1iT - 43T^{2} \) |
| 47 | \( 1 + 11.1iT - 47T^{2} \) |
| 53 | \( 1 - 2.26T + 53T^{2} \) |
| 59 | \( 1 - 4.61T + 59T^{2} \) |
| 61 | \( 1 - 11.8iT - 61T^{2} \) |
| 67 | \( 1 - 6.90iT - 67T^{2} \) |
| 71 | \( 1 + 2.63iT - 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 - 8.01T + 79T^{2} \) |
| 83 | \( 1 - 3.20iT - 83T^{2} \) |
| 89 | \( 1 + 17.8T + 89T^{2} \) |
| 97 | \( 1 + 8.68T + 97T^{2} \) |
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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
−8.327457864955381261680302174996, −7.38620060802062622652795677142, −6.97962167907580436905124872397, −6.12570582086976718794084546004, −5.29125089740733730397712165848, −4.24822064416132024000975287154, −3.18201809179626461189918601469, −2.58582757916499255223281476691, −1.00960643775017485623493210670, −0.094276112353557039839528094392,
1.73712502031506145192815842108, 2.33781288695137898802605932851, 3.57491806773274905914523954273, 4.35670623001439732940723105477, 5.63749653713096326543381933131, 6.14447520481658555671742602153, 6.93878577293575355081781582333, 7.78183728373465723732927161167, 8.373565542811946055134514175359, 9.160077437288898496421940837080
