L(s) = 1 | + (1.24 − 0.671i)2-s + 3-s + (1.09 − 1.67i)4-s + (1.24 − 0.671i)6-s + 4.68i·7-s + (0.244 − 2.81i)8-s + 9-s − 2.29i·11-s + (1.09 − 1.67i)12-s + 4.97·13-s + (3.14 + 5.83i)14-s + (−1.58 − 3.67i)16-s − 2.97i·17-s + (1.24 − 0.671i)18-s + 2.68i·19-s + ⋯ |
L(s) = 1 | + (0.880 − 0.474i)2-s + 0.577·3-s + (0.549 − 0.835i)4-s + (0.508 − 0.274i)6-s + 1.77i·7-s + (0.0864 − 0.996i)8-s + 0.333·9-s − 0.691i·11-s + (0.317 − 0.482i)12-s + 1.38·13-s + (0.840 + 1.55i)14-s + (−0.396 − 0.917i)16-s − 0.722i·17-s + (0.293 − 0.158i)18-s + 0.616i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.91328 - 0.822374i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.91328 - 0.822374i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.24 + 0.671i)T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4.68iT - 7T^{2} \) |
| 11 | \( 1 + 2.29iT - 11T^{2} \) |
| 13 | \( 1 - 4.97T + 13T^{2} \) |
| 17 | \( 1 + 2.97iT - 17T^{2} \) |
| 19 | \( 1 - 2.68iT - 19T^{2} \) |
| 23 | \( 1 + 2.68iT - 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 6.97T + 31T^{2} \) |
| 37 | \( 1 - 4.39T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + 9.37T + 43T^{2} \) |
| 47 | \( 1 - 7.27iT - 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 1.70iT - 59T^{2} \) |
| 61 | \( 1 + 4.58iT - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 0.585T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 1.02T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 3.37T + 89T^{2} \) |
| 97 | \( 1 + 3.95iT - 97T^{2} \) |
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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
−10.84955405386415126659492257479, −9.684044360727980832230064250618, −8.879883947830220902457173992825, −8.189595967031247087359708588524, −6.65554910077042468251844151102, −5.85474671745086958442085103907, −5.08069673183854232950890688261, −3.64469976760805615710973763927, −2.86510314507904884962965152604, −1.70706145026574424906716649019,
1.69841650150167190005975056670, 3.49765089720432791333809877007, 3.94742907484286548935946397024, 5.01823876768522536197359845525, 6.41273348984488609322165730348, 7.10185490142539487624984042017, 7.86506811036578009990644802577, 8.715472323445342519948238649533, 10.01370651112997052809573112429, 10.80862653420548213591625336159