L(s) = 1 | + (−0.635 + 1.26i)2-s + (−1.19 − 1.60i)4-s + (0.707 + 0.707i)5-s − 1.41i·7-s + (2.78 − 0.484i)8-s + (−1.34 + 0.443i)10-s + (−1.11 − 1.11i)11-s + (−0.271 + 0.271i)13-s + (1.78 + 0.898i)14-s + (−1.15 + 3.82i)16-s − 0.744·17-s + (5.21 − 5.21i)19-s + (0.292 − 1.97i)20-s + (2.11 − 0.698i)22-s − 4.76i·23-s + ⋯ |
L(s) = 1 | + (−0.449 + 0.893i)2-s + (−0.595 − 0.803i)4-s + (0.316 + 0.316i)5-s − 0.534i·7-s + (0.985 − 0.171i)8-s + (−0.424 + 0.140i)10-s + (−0.335 − 0.335i)11-s + (−0.0752 + 0.0752i)13-s + (0.477 + 0.240i)14-s + (−0.289 + 0.957i)16-s − 0.180·17-s + (1.19 − 1.19i)19-s + (0.0654 − 0.442i)20-s + (0.450 − 0.148i)22-s − 0.994i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13523 + 0.0560971i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13523 + 0.0560971i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.635 - 1.26i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 + 1.41iT - 7T^{2} \) |
| 11 | \( 1 + (1.11 + 1.11i)T + 11iT^{2} \) |
| 13 | \( 1 + (0.271 - 0.271i)T - 13iT^{2} \) |
| 17 | \( 1 + 0.744T + 17T^{2} \) |
| 19 | \( 1 + (-5.21 + 5.21i)T - 19iT^{2} \) |
| 23 | \( 1 + 4.76iT - 23T^{2} \) |
| 29 | \( 1 + (-1.21 + 1.21i)T - 29iT^{2} \) |
| 31 | \( 1 - 7.75T + 31T^{2} \) |
| 37 | \( 1 + (5.32 + 5.32i)T + 37iT^{2} \) |
| 41 | \( 1 + 7.33iT - 41T^{2} \) |
| 43 | \( 1 + (-6.78 - 6.78i)T + 43iT^{2} \) |
| 47 | \( 1 - 0.735T + 47T^{2} \) |
| 53 | \( 1 + (-9.55 - 9.55i)T + 53iT^{2} \) |
| 59 | \( 1 + (1.62 + 1.62i)T + 59iT^{2} \) |
| 61 | \( 1 + (5.70 - 5.70i)T - 61iT^{2} \) |
| 67 | \( 1 + (-5.59 + 5.59i)T - 67iT^{2} \) |
| 71 | \( 1 + 8.60iT - 71T^{2} \) |
| 73 | \( 1 - 4.28iT - 73T^{2} \) |
| 79 | \( 1 + 1.01T + 79T^{2} \) |
| 83 | \( 1 + (1.68 - 1.68i)T - 83iT^{2} \) |
| 89 | \( 1 + 10.3iT - 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−10.40555080016747501417422202967, −9.404356094729947979121712810251, −8.726451792788410176964215020281, −7.65547985414276719623168564173, −7.02436658543934524967903079745, −6.14353484465609318531529284521, −5.18480604389193523423019580779, −4.20859546502324061033422252014, −2.63643136114201967280396452464, −0.797592002734954829268969664258,
1.31693599469307774864356098127, 2.53026191678520538622918158296, 3.63061979040357615351536242724, 4.87693928807789818072246620132, 5.71425076329546817710333240756, 7.15006353402501026188359565280, 8.086523850420905769529366651604, 8.788018160250684166158190108359, 9.825058571543425649265073004241, 10.09566188170960289904861595011