| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s − 4·11-s + 12-s + 2·13-s − 14-s + 16-s − 17-s + 18-s − 21-s − 4·22-s + 4·23-s + 24-s + 2·26-s + 27-s − 28-s − 2·29-s + 32-s − 4·33-s − 34-s + 36-s − 2·37-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 − 1.20·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.218·21-s − 0.852·22-s + 0.834·23-s + 0.204·24-s + 0.392·26-s + 0.192·27-s − 0.188·28-s − 0.371·29-s + 0.176·32-s − 0.696·33-s − 0.171·34-s + 1/6·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.99499542083295, −15.31446975995366, −15.11190703111289, −14.46691722995134, −13.71112356023966, −13.28925387984094, −13.07499452156843, −12.39207206952015, −11.72939165854881, −11.07466979300861, −10.55668711902106, −9.995065695986033, −9.367804750337381, −8.575822157232651, −8.164940701810919, −7.451430568887362, −6.824076737641144, −6.312461309134753, −5.387763253978932, −5.032041238668900, −4.196962169457596, −3.410590337197996, −2.991735336857925, −2.209356704020345, −1.373161900627025, 0,
1.373161900627025, 2.209356704020345, 2.991735336857925, 3.410590337197996, 4.196962169457596, 5.032041238668900, 5.387763253978932, 6.312461309134753, 6.824076737641144, 7.451430568887362, 8.164940701810919, 8.575822157232651, 9.367804750337381, 9.995065695986033, 10.55668711902106, 11.07466979300861, 11.72939165854881, 12.39207206952015, 13.07499452156843, 13.28925387984094, 13.71112356023966, 14.46691722995134, 15.11190703111289, 15.31446975995366, 15.99499542083295
