L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s − 2·9-s + 10-s + 12-s − 2·13-s − 15-s + 16-s + 5·17-s + 2·18-s + 4·19-s − 20-s + 6·23-s − 24-s − 4·25-s + 2·26-s − 5·27-s − 9·29-s + 30-s − 3·31-s − 32-s − 5·34-s − 2·36-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 − 2/3·9-s + 0.316·10-s + 0.288·12-s − 0.554·13-s − 0.258·15-s + 1/4·16-s + 1.21·17-s + 0.471·18-s + 0.917·19-s − 0.223·20-s + 1.25·23-s − 0.204·24-s − 4/5·25-s + 0.392·26-s − 0.962·27-s − 1.67·29-s + 0.182·30-s − 0.538·31-s − 0.176·32-s − 0.857·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−8.103708430326373706384192857771, −7.40622866776395561714576772584, −7.10847700689501933078392137011, −5.64666303194085933243943394791, −5.42471489426608259988012070037, −3.96534924163346245201203022127, −3.26336529204343825908177421930, −2.50599759115468772534756961383, −1.35572225109890639211902170084, 0,
1.35572225109890639211902170084, 2.50599759115468772534756961383, 3.26336529204343825908177421930, 3.96534924163346245201203022127, 5.42471489426608259988012070037, 5.64666303194085933243943394791, 7.10847700689501933078392137011, 7.40622866776395561714576772584, 8.103708430326373706384192857771