L(s) = 1 | − 2-s + 4-s − 5-s − 2.95i·7-s − 8-s + 10-s − 3.40·11-s − 2.36i·13-s + 2.95i·14-s + 16-s + 3.78i·17-s + 6.74·19-s − 20-s + 3.40·22-s − 2.59i·23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 1.11i·7-s − 0.353·8-s + 0.316·10-s − 1.02·11-s − 0.655i·13-s + 0.790i·14-s + 0.250·16-s + 0.919i·17-s + 1.54·19-s − 0.223·20-s + 0.725·22-s − 0.540i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7198335035\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7198335035\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 67 | \( 1 + (8.17 + 0.307i)T \) |
good | 7 | \( 1 + 2.95iT - 7T^{2} \) |
| 11 | \( 1 + 3.40T + 11T^{2} \) |
| 13 | \( 1 + 2.36iT - 13T^{2} \) |
| 17 | \( 1 - 3.78iT - 17T^{2} \) |
| 19 | \( 1 - 6.74T + 19T^{2} \) |
| 23 | \( 1 + 2.59iT - 23T^{2} \) |
| 29 | \( 1 - 0.879iT - 29T^{2} \) |
| 31 | \( 1 - 9.80iT - 31T^{2} \) |
| 37 | \( 1 + 7.74T + 37T^{2} \) |
| 41 | \( 1 + 3.03T + 41T^{2} \) |
| 43 | \( 1 + 0.700iT - 43T^{2} \) |
| 47 | \( 1 - 13.3iT - 47T^{2} \) |
| 53 | \( 1 + 8.97T + 53T^{2} \) |
| 59 | \( 1 + 10.5iT - 59T^{2} \) |
| 61 | \( 1 + 10.1iT - 61T^{2} \) |
| 71 | \( 1 + 7.98iT - 71T^{2} \) |
| 73 | \( 1 + 2.35T + 73T^{2} \) |
| 79 | \( 1 + 2.08iT - 79T^{2} \) |
| 83 | \( 1 - 4.90iT - 83T^{2} \) |
| 89 | \( 1 + 2.96iT - 89T^{2} \) |
| 97 | \( 1 + 1.73iT - 97T^{2} \) |
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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
−8.008574483801182104649839788462, −7.65584152284374797599308993922, −7.00607421648698390468025230071, −6.24876764506122799597939408209, −5.24112946812940686402369560181, −4.66447236644298426285906717230, −3.40310214850572400226113666785, −3.14329825485219705240880928682, −1.71620640834698480985936885114, −0.78756327088464240788640374879,
0.31394156508993448726578767133, 1.69623675052427190134206073229, 2.61525695630692991361927133052, 3.21414671642405462562962297770, 4.35401552405759270720667905101, 5.46992171029138773557048799939, 5.57666124404037732577745216869, 6.83085274868349016871940966503, 7.38919364572154328603710372332, 7.972086363348436475350250388508