L(s) = 1 | + 2-s − 2·3-s + 4-s − 5-s − 2·6-s − 7-s + 8-s + 9-s − 10-s − 6·11-s − 2·12-s + 5·13-s − 14-s + 2·15-s + 16-s − 6·17-s + 18-s − 7·19-s − 20-s + 2·21-s − 6·22-s − 6·23-s − 2·24-s + 25-s + 5·26-s + 4·27-s − 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.80·11-s − 0.577·12-s + 1.38·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 1.60·19-s − 0.223·20-s + 0.436·21-s − 1.27·22-s − 1.25·23-s − 0.408·24-s + 1/5·25-s + 0.980·26-s + 0.769·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−10.73802668415093071200389163520, −10.50382842602194265022166283510, −8.725648395491444250898297690863, −7.82830444847008932868735016407, −6.46215731525971101857740013155, −6.02408712028264076876486984857, −4.90814848007890474751744003516, −3.97591565724208160530804100347, −2.44719165435151688936148894351, 0,
2.44719165435151688936148894351, 3.97591565724208160530804100347, 4.90814848007890474751744003516, 6.02408712028264076876486984857, 6.46215731525971101857740013155, 7.82830444847008932868735016407, 8.725648395491444250898297690863, 10.50382842602194265022166283510, 10.73802668415093071200389163520