| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s + 9-s − 10-s + 11-s + 12-s + 3·13-s − 14-s − 15-s + 16-s + 18-s − 6·19-s − 20-s − 21-s + 22-s + 23-s + 24-s − 4·25-s + 3·26-s + 27-s − 28-s − 9·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.832·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.235·18-s − 1.37·19-s − 0.223·20-s − 0.218·21-s + 0.213·22-s + 0.208·23-s + 0.204·24-s − 4/5·25-s + 0.588·26-s + 0.192·27-s − 0.188·28-s − 1.67·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10626 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10626 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−16.65128429503049, −16.19475157568872, −15.37049526922833, −15.25051701678476, −14.54224928656981, −14.01429006568290, −13.31115345682586, −12.92080460092010, −12.46845926886284, −11.58264983973067, −11.18173114087450, −10.54413842501377, −9.820159687439307, −9.049070705234170, −8.619572484335568, −7.752462164361339, −7.352341942382106, −6.406353234333195, −6.079745345441717, −5.147237227317780, −4.269356198637971, −3.758638567257546, −3.262235501642129, −2.223210858398100, −1.525599916130401, 0,
1.525599916130401, 2.223210858398100, 3.262235501642129, 3.758638567257546, 4.269356198637971, 5.147237227317780, 6.079745345441717, 6.406353234333195, 7.352341942382106, 7.752462164361339, 8.619572484335568, 9.049070705234170, 9.820159687439307, 10.54413842501377, 11.18173114087450, 11.58264983973067, 12.46845926886284, 12.92080460092010, 13.31115345682586, 14.01429006568290, 14.54224928656981, 15.25051701678476, 15.37049526922833, 16.19475157568872, 16.65128429503049
