| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 3·7-s + 8-s + 9-s − 11-s + 12-s + 13-s − 3·14-s + 16-s + 17-s + 18-s + 4·19-s − 3·21-s − 22-s + 2·23-s + 24-s + 26-s + 27-s − 3·28-s + 6·29-s − 4·31-s + 32-s − 33-s + 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 0.277·13-s − 0.801·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.654·21-s − 0.213·22-s + 0.417·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s − 0.566·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.174·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 11 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.73802238373505, −12.47427967648449, −11.94546661041538, −11.49713024799174, −10.97342171689930, −10.34355078722648, −10.09474653378793, −9.643665940811403, −9.107408850423103, −8.617129201413128, −8.231098997212275, −7.500724410373329, −7.176778966282352, −6.792421612358544, −6.153679838490309, −5.840011965243366, −5.174199768114249, −4.759115284849914, −4.196289563780400, −3.415605317288506, −3.258785129724213, −2.927468044893468, −2.129455522418083, −1.572325164759793, −0.8563454825175984, 0,
0.8563454825175984, 1.572325164759793, 2.129455522418083, 2.927468044893468, 3.258785129724213, 3.415605317288506, 4.196289563780400, 4.759115284849914, 5.174199768114249, 5.840011965243366, 6.153679838490309, 6.792421612358544, 7.176778966282352, 7.500724410373329, 8.231098997212275, 8.617129201413128, 9.107408850423103, 9.643665940811403, 10.09474653378793, 10.34355078722648, 10.97342171689930, 11.49713024799174, 11.94546661041538, 12.47427967648449, 12.73802238373505
