L(s) = 1 | + (0.5 − 0.866i)5-s + (0.939 + 0.342i)9-s + (0.592 + 1.62i)13-s + (−0.439 + 1.20i)17-s + (−0.499 − 0.866i)25-s + (0.939 + 1.62i)29-s + (−0.939 + 0.342i)37-s + (1.43 − 0.524i)41-s + (0.766 − 0.642i)45-s + (−0.173 − 0.984i)49-s + (−1.11 − 1.32i)53-s + (−1.76 + 0.642i)61-s + (1.70 + 0.300i)65-s − 1.73i·73-s + (0.766 + 0.642i)81-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)5-s + (0.939 + 0.342i)9-s + (0.592 + 1.62i)13-s + (−0.439 + 1.20i)17-s + (−0.499 − 0.866i)25-s + (0.939 + 1.62i)29-s + (−0.939 + 0.342i)37-s + (1.43 − 0.524i)41-s + (0.766 − 0.642i)45-s + (−0.173 − 0.984i)49-s + (−1.11 − 1.32i)53-s + (−1.76 + 0.642i)61-s + (1.70 + 0.300i)65-s − 1.73i·73-s + (0.766 + 0.642i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.481864863\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.481864863\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.939 - 0.342i)T \) |
good | 3 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.592 - 1.62i)T + (-0.766 + 0.642i)T^{2} \) |
| 17 | \( 1 + (0.439 - 1.20i)T + (-0.766 - 0.642i)T^{2} \) |
| 19 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-1.43 + 0.524i)T + (0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (1.11 + 1.32i)T + (-0.173 + 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \) |
| 67 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 73 | \( 1 + 1.73iT - T^{2} \) |
| 79 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 83 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 89 | \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 97 | \( 1 + (0.592 + 0.342i)T + (0.5 + 0.866i)T^{2} \) |
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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
−8.848783161453405483665834183237, −8.480794721828214663694714985375, −7.39705352333432086953325069025, −6.60785856572040316126483361924, −6.02169852799747054102831063600, −4.86490024805418520897403517120, −4.44381553768921732595813673864, −3.52770570051339012888827305077, −1.92479720447447357976641160795, −1.49358667221553987625071396665,
1.06162985192457472635288034548, 2.47251635465140625155527843455, 3.13365276779110987950459235299, 4.14452673835279198071711663017, 5.08267186395811083334213609417, 6.06396291775232058139535177913, 6.49753384360638582715103577026, 7.53138318362009678337159091987, 7.86283374976500743090629282043, 9.113088016783039748765909611855