| L(s) = 1 | − 2-s − 4-s + 7-s + 3·8-s − 4·13-s − 14-s − 16-s − 2·17-s + 4·26-s − 28-s + 8·29-s − 4·31-s − 5·32-s + 2·34-s − 8·37-s + 4·41-s − 8·43-s + 12·47-s + 49-s + 4·52-s − 6·53-s + 3·56-s − 8·58-s + 8·59-s + 10·61-s + 4·62-s + 7·64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s − 1.10·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.784·26-s − 0.188·28-s + 1.48·29-s − 0.718·31-s − 0.883·32-s + 0.342·34-s − 1.31·37-s + 0.624·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-s + 0.554·52-s − 0.824·53-s + 0.400·56-s − 1.05·58-s + 1.04·59-s + 1.28·61-s + 0.508·62-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−8.836977164289154374877069968146, −8.515435984898579188140145667992, −7.46935627665195467124923832071, −6.96731122673944982386764065211, −5.62709236482279603070071058958, −4.80666074841933533046073813928, −4.10234838538757672649423052413, −2.70046968569181803421525831736, −1.47198786062397073562548092470, 0,
1.47198786062397073562548092470, 2.70046968569181803421525831736, 4.10234838538757672649423052413, 4.80666074841933533046073813928, 5.62709236482279603070071058958, 6.96731122673944982386764065211, 7.46935627665195467124923832071, 8.515435984898579188140145667992, 8.836977164289154374877069968146
