L(s) = 1 | + (0.984 − 0.173i)2-s + (0.939 − 0.342i)4-s + (−0.866 − 0.5i)5-s + (0.866 − 0.5i)8-s + (0.342 − 0.939i)9-s + (−0.939 − 0.342i)10-s + (0.592 + 1.62i)13-s + (0.766 − 0.642i)16-s + (−1.43 − 0.524i)17-s + (0.173 − 0.984i)18-s + (−0.984 − 0.173i)20-s + (0.499 + 0.866i)25-s + (0.866 + 1.5i)26-s + (−0.816 − 0.218i)29-s + (0.642 − 0.766i)32-s + ⋯ |
L(s) = 1 | + (0.984 − 0.173i)2-s + (0.939 − 0.342i)4-s + (−0.866 − 0.5i)5-s + (0.866 − 0.5i)8-s + (0.342 − 0.939i)9-s + (−0.939 − 0.342i)10-s + (0.592 + 1.62i)13-s + (0.766 − 0.642i)16-s + (−1.43 − 0.524i)17-s + (0.173 − 0.984i)18-s + (−0.984 − 0.173i)20-s + (0.499 + 0.866i)25-s + (0.866 + 1.5i)26-s + (−0.816 − 0.218i)29-s + (0.642 − 0.766i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.542607754\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.542607754\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.984 + 0.173i)T \) |
| 5 | \( 1 + (0.866 + 0.5i)T \) |
| 37 | \( 1 + (-0.342 - 0.939i)T \) |
good | 3 | \( 1 + (-0.342 + 0.939i)T^{2} \) |
| 7 | \( 1 + (0.984 - 0.173i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.592 - 1.62i)T + (-0.766 + 0.642i)T^{2} \) |
| 17 | \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \) |
| 19 | \( 1 + (-0.342 + 0.939i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.816 + 0.218i)T + (0.866 + 0.5i)T^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (-0.439 - 1.20i)T + (-0.766 + 0.642i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (0.168 - 1.92i)T + (-0.984 - 0.173i)T^{2} \) |
| 59 | \( 1 + (-0.984 - 0.173i)T^{2} \) |
| 61 | \( 1 + (1.64 + 0.766i)T + (0.642 + 0.766i)T^{2} \) |
| 67 | \( 1 + (-0.984 + 0.173i)T^{2} \) |
| 71 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 73 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
| 79 | \( 1 + (0.984 - 0.173i)T^{2} \) |
| 83 | \( 1 + (-0.642 + 0.766i)T^{2} \) |
| 89 | \( 1 + (-0.173 + 1.98i)T + (-0.984 - 0.173i)T^{2} \) |
| 97 | \( 1 + (0.342 - 0.592i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−10.93685831820392977520878477882, −9.511640512286199479088442674560, −8.940042802451927332511392618042, −7.66942698320789075568262022680, −6.76219825226390380544567842302, −6.15333947361185046594758580438, −4.57096043062824988502342279596, −4.29578448807597526264068318682, −3.16738692510039186285834946474, −1.57231095733345363035232617912,
2.19702367912220988542991785592, 3.36184895739625839583905019400, 4.22046012538914399754228271971, 5.22087214487291229796573507897, 6.19265706003014955362726252774, 7.20810738661568882024042344905, 7.84746669332765959047675377519, 8.606775238626609185708893790634, 10.30122719400437131432966773205, 10.91514695417948754677744863679