| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 4·11-s − 12-s − 2·13-s + 15-s + 16-s + 18-s + 4·19-s − 20-s − 4·22-s + 23-s − 24-s + 25-s − 2·26-s − 27-s + 2·29-s + 30-s + 32-s + 4·33-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-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 + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.852·22-s + 0.208·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.371·29-s + 0.182·30-s + 0.176·32-s + 0.696·33-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 199410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−13.26550674129082, −12.87904356205177, −12.29810445912132, −11.90716189629261, −11.57469757607543, −11.07754877910831, −10.56615106647587, −9.982902908493246, −9.924033426487821, −8.979110718362926, −8.485355196598585, −7.833342875841654, −7.517941116738411, −7.063236947820992, −6.480527418777256, −5.992616086949606, −5.337342314054778, −4.908812405177605, −4.787176013340769, −3.902370300902237, −3.368410392313957, −2.882044370065754, −2.262474221932297, −1.557382284495767, −0.7338263799731364, 0,
0.7338263799731364, 1.557382284495767, 2.262474221932297, 2.882044370065754, 3.368410392313957, 3.902370300902237, 4.787176013340769, 4.908812405177605, 5.337342314054778, 5.992616086949606, 6.480527418777256, 7.063236947820992, 7.517941116738411, 7.833342875841654, 8.485355196598585, 8.979110718362926, 9.924033426487821, 9.982902908493246, 10.56615106647587, 11.07754877910831, 11.57469757607543, 11.90716189629261, 12.29810445912132, 12.87904356205177, 13.26550674129082
