| L(s) = 1 | − 2-s − 3-s + 4-s + 0.326·5-s + 6-s − 1.59·7-s − 8-s + 9-s − 0.326·10-s − 2.21·11-s − 12-s + 2.49·13-s + 1.59·14-s − 0.326·15-s + 16-s + 6.46·17-s − 18-s − 7.66·19-s + 0.326·20-s + 1.59·21-s + 2.21·22-s + 2.91·23-s + 24-s − 4.89·25-s − 2.49·26-s − 27-s − 1.59·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.146·5-s + 0.408·6-s − 0.603·7-s − 0.353·8-s + 0.333·9-s − 0.103·10-s − 0.669·11-s − 0.288·12-s + 0.690·13-s + 0.426·14-s − 0.0842·15-s + 0.250·16-s + 1.56·17-s − 0.235·18-s − 1.75·19-s + 0.0730·20-s + 0.348·21-s + 0.473·22-s + 0.607·23-s + 0.204·24-s − 0.978·25-s − 0.488·26-s − 0.192·27-s − 0.301·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 - 0.326T + 5T^{2} \) |
| 7 | \( 1 + 1.59T + 7T^{2} \) |
| 11 | \( 1 + 2.21T + 11T^{2} \) |
| 13 | \( 1 - 2.49T + 13T^{2} \) |
| 17 | \( 1 - 6.46T + 17T^{2} \) |
| 19 | \( 1 + 7.66T + 19T^{2} \) |
| 23 | \( 1 - 2.91T + 23T^{2} \) |
| 29 | \( 1 + 4.18T + 29T^{2} \) |
| 31 | \( 1 - 5.66T + 31T^{2} \) |
| 37 | \( 1 + 2.05T + 37T^{2} \) |
| 41 | \( 1 - 12.2T + 41T^{2} \) |
| 43 | \( 1 + 8.89T + 43T^{2} \) |
| 47 | \( 1 + 8.26T + 47T^{2} \) |
| 53 | \( 1 + 6.20T + 53T^{2} \) |
| 59 | \( 1 + 0.283T + 59T^{2} \) |
| 61 | \( 1 + 2.94T + 61T^{2} \) |
| 67 | \( 1 + 5.67T + 67T^{2} \) |
| 71 | \( 1 - 4.12T + 71T^{2} \) |
| 73 | \( 1 - 7.69T + 73T^{2} \) |
| 79 | \( 1 + 0.116T + 79T^{2} \) |
| 83 | \( 1 + 7.08T + 83T^{2} \) |
| 89 | \( 1 - 0.513T + 89T^{2} \) |
| 97 | \( 1 + 6.38T + 97T^{2} \) |
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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
−9.415982917934371246851115845873, −8.299396981918821509374762295242, −7.76442035004950498182218434377, −6.62974995463474002646012617052, −6.08909485588767440162569769402, −5.21752186008420517175535545674, −3.93312396850202973857380928848, −2.84122260867046052460528311042, −1.49242435063325155810800722200, 0,
1.49242435063325155810800722200, 2.84122260867046052460528311042, 3.93312396850202973857380928848, 5.21752186008420517175535545674, 6.08909485588767440162569769402, 6.62974995463474002646012617052, 7.76442035004950498182218434377, 8.299396981918821509374762295242, 9.415982917934371246851115845873
