| L(s) = 1 | − 2-s − 3-s + 4-s − 4·5-s + 6-s + 7-s − 8-s + 9-s + 4·10-s − 3·11-s − 12-s + 13-s − 14-s + 4·15-s + 16-s − 3·17-s − 18-s − 8·19-s − 4·20-s − 21-s + 3·22-s − 6·23-s + 24-s + 11·25-s − 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.904·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 1.03·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 1.83·19-s − 0.894·20-s − 0.218·21-s + 0.639·22-s − 1.25·23-s + 0.204·24-s + 11/5·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.21094996713725333076978667776, −6.67139079504067841107749348025, −5.91607941836104482632201668555, −4.84661324687243618624883582403, −4.34184806672523524088911538729, −3.61370170728669113635507924612, −2.57921501210518350813689961886, −1.55929325184777123326212159667, 0, 0,
1.55929325184777123326212159667, 2.57921501210518350813689961886, 3.61370170728669113635507924612, 4.34184806672523524088911538729, 4.84661324687243618624883582403, 5.91607941836104482632201668555, 6.67139079504067841107749348025, 7.21094996713725333076978667776
