| L(s) = 1 | − 2-s + 3-s − 4-s + 5-s − 6-s + 3·8-s + 9-s − 10-s + 4·11-s − 12-s + 2·13-s + 15-s − 16-s − 2·17-s − 18-s + 4·19-s − 20-s − 4·22-s + 3·24-s + 25-s − 2·26-s + 27-s + 2·29-s − 30-s − 5·32-s + 4·33-s + 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.554·13-s + 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.852·22-s + 0.612·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.371·29-s − 0.182·30-s − 0.883·32-s + 0.696·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.293548531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.293548531\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−16.16794171727104, −15.72933349060585, −14.76716257784066, −14.43645436858361, −13.90988221229338, −13.35602697588368, −12.98213989062862, −12.17540651757322, −11.45187923155449, −10.92709379082204, −10.16335858261843, −9.623815085347354, −9.223978130094169, −8.752114695372547, −8.171383858653182, −7.488359096159968, −6.893996779456610, −6.127358842906413, −5.465212030158474, −4.442925271805524, −4.122305666468883, −3.241097610702859, −2.333661075480660, −1.383630623374400, −0.8327825402824426,
0.8327825402824426, 1.383630623374400, 2.333661075480660, 3.241097610702859, 4.122305666468883, 4.442925271805524, 5.465212030158474, 6.127358842906413, 6.893996779456610, 7.488359096159968, 8.171383858653182, 8.752114695372547, 9.223978130094169, 9.623815085347354, 10.16335858261843, 10.92709379082204, 11.45187923155449, 12.17540651757322, 12.98213989062862, 13.35602697588368, 13.90988221229338, 14.43645436858361, 14.76716257784066, 15.72933349060585, 16.16794171727104
