| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s − 11-s + 12-s − 4·13-s − 14-s + 16-s − 18-s + 2·19-s + 21-s + 22-s + 6·23-s − 24-s + 4·26-s + 27-s + 28-s − 2·29-s − 2·31-s − 32-s − 33-s + 36-s − 6·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.458·19-s + 0.218·21-s + 0.213·22-s + 1.25·23-s − 0.204·24-s + 0.784·26-s + 0.192·27-s + 0.188·28-s − 0.371·29-s − 0.359·31-s − 0.176·32-s − 0.174·33-s + 1/6·36-s − 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−16.88937124782685, −16.14267684308547, −15.55801103791690, −14.99060712291371, −14.65877269688642, −13.90823731425770, −13.34953049475332, −12.68309182978296, −12.00339579648006, −11.60008562914948, −10.65762869467971, −10.37002501145792, −9.646881857859091, −8.994880723545098, −8.657117009576716, −7.820340727000878, −7.251127020070215, −6.994060724711973, −5.869805359091416, −5.157934395708951, −4.548323153177455, −3.483102867321374, −2.838474414726412, −2.070020084096770, −1.249242436581640, 0,
1.249242436581640, 2.070020084096770, 2.838474414726412, 3.483102867321374, 4.548323153177455, 5.157934395708951, 5.869805359091416, 6.994060724711973, 7.251127020070215, 7.820340727000878, 8.657117009576716, 8.994880723545098, 9.646881857859091, 10.37002501145792, 10.65762869467971, 11.60008562914948, 12.00339579648006, 12.68309182978296, 13.34953049475332, 13.90823731425770, 14.65877269688642, 14.99060712291371, 15.55801103791690, 16.14267684308547, 16.88937124782685
