| L(s) = 1 | + 2·5-s − 8·7-s − 9-s − 6·13-s − 25-s + 4·29-s − 16·35-s − 4·37-s − 2·45-s − 24·47-s + 34·49-s + 4·61-s + 8·63-s − 12·65-s + 16·67-s − 28·73-s + 81-s + 48·91-s − 20·97-s + 12·101-s + 6·117-s + 22·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 3.02·7-s − 1/3·9-s − 1.66·13-s − 1/5·25-s + 0.742·29-s − 2.70·35-s − 0.657·37-s − 0.298·45-s − 3.50·47-s + 34/7·49-s + 0.512·61-s + 1.00·63-s − 1.48·65-s + 1.95·67-s − 3.27·73-s + 1/9·81-s + 5.03·91-s − 2.03·97-s + 1.19·101-s + 0.554·117-s + 2·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.426000280155452982522216947154, −8.987390352523574423172333012150, −8.586434480806173102711115419567, −8.114602975625271709032564743681, −7.35793787528840869435835401912, −7.24346473184260868321218001244, −6.58557888072885847869487120757, −6.45710434808406877037042938503, −6.14922582812079074347920442029, −5.67128770967748365674011819524, −5.00143772800929262410841532151, −4.92706777708603127367494142276, −3.94749762617663899327923007641, −3.57975548120889375946505245936, −3.03740593878156490302997675854, −2.70132627457938943608826921633, −2.30782008644998108200790310259, −1.39444465766445501081193218156, 0, 0,
1.39444465766445501081193218156, 2.30782008644998108200790310259, 2.70132627457938943608826921633, 3.03740593878156490302997675854, 3.57975548120889375946505245936, 3.94749762617663899327923007641, 4.92706777708603127367494142276, 5.00143772800929262410841532151, 5.67128770967748365674011819524, 6.14922582812079074347920442029, 6.45710434808406877037042938503, 6.58557888072885847869487120757, 7.24346473184260868321218001244, 7.35793787528840869435835401912, 8.114602975625271709032564743681, 8.586434480806173102711115419567, 8.987390352523574423172333012150, 9.426000280155452982522216947154
