L(s) = 1 | − 2-s − 2·3-s + 4-s − 3·5-s + 2·6-s − 4·7-s − 8-s + 9-s + 3·10-s − 6·11-s − 2·12-s − 5·13-s + 4·14-s + 6·15-s + 16-s − 5·17-s − 18-s + 19-s − 3·20-s + 8·21-s + 6·22-s − 9·23-s + 2·24-s + 4·25-s + 5·26-s + 4·27-s − 4·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.34·5-s + 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 1.80·11-s − 0.577·12-s − 1.38·13-s + 1.06·14-s + 1.54·15-s + 1/4·16-s − 1.21·17-s − 0.235·18-s + 0.229·19-s − 0.670·20-s + 1.74·21-s + 1.27·22-s − 1.87·23-s + 0.408·24-s + 4/5·25-s + 0.980·26-s + 0.769·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18562 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18562 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 9281 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−16.44984905247392, −16.05717430726503, −15.77351519940404, −15.37917942213979, −14.62486090360148, −13.70909255181529, −12.99125659125134, −12.51785819972882, −12.04496565175918, −11.73846784938250, −10.78591469976345, −10.64723676512456, −9.984050681033143, −9.474233419395539, −8.616988548785839, −7.997544945560637, −7.440993418419587, −6.972405170288869, −6.315362930154198, −5.705143793886162, −4.953573741704914, −4.395151267663604, −3.336989229039679, −2.866845554197706, −1.941489667876912, 0, 0, 0,
1.941489667876912, 2.866845554197706, 3.336989229039679, 4.395151267663604, 4.953573741704914, 5.705143793886162, 6.315362930154198, 6.972405170288869, 7.440993418419587, 7.997544945560637, 8.616988548785839, 9.474233419395539, 9.984050681033143, 10.64723676512456, 10.78591469976345, 11.73846784938250, 12.04496565175918, 12.51785819972882, 12.99125659125134, 13.70909255181529, 14.62486090360148, 15.37917942213979, 15.77351519940404, 16.05717430726503, 16.44984905247392