| L(s) = 1 | − 2·5-s + 6·13-s − 8·17-s − 7·25-s + 2·29-s + 16·37-s − 10·41-s − 11·49-s − 16·53-s + 14·61-s − 12·65-s − 24·73-s + 16·85-s + 8·89-s − 6·97-s − 26·101-s + 2·113-s − 19·121-s + 26·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.66·13-s − 1.94·17-s − 7/5·25-s + 0.371·29-s + 2.63·37-s − 1.56·41-s − 1.57·49-s − 2.19·53-s + 1.79·61-s − 1.48·65-s − 2.80·73-s + 1.73·85-s + 0.847·89-s − 0.609·97-s − 2.58·101-s + 0.188·113-s − 1.72·121-s + 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.332·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 131 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 91 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
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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
−8.017079140951883488070335008157, −7.87952044240999256766427664650, −7.35405543094406606722650232925, −6.87439164003596768971370692637, −6.55697203242958446216014355797, −6.33797914767129957760086287711, −5.88096134532344983890968030900, −5.68785511731878536963599555958, −4.89640991614013405661000721394, −4.69051408731632134830622147168, −4.17060095760202355627080048597, −4.09011670481825052910150333744, −3.42151429483412598455337828239, −3.32985615785838910909568164509, −2.48022858347745485693222102074, −2.30903502624702498703141712776, −1.40070458817986799643669663738, −1.27928161018926854390990233596, 0, 0,
1.27928161018926854390990233596, 1.40070458817986799643669663738, 2.30903502624702498703141712776, 2.48022858347745485693222102074, 3.32985615785838910909568164509, 3.42151429483412598455337828239, 4.09011670481825052910150333744, 4.17060095760202355627080048597, 4.69051408731632134830622147168, 4.89640991614013405661000721394, 5.68785511731878536963599555958, 5.88096134532344983890968030900, 6.33797914767129957760086287711, 6.55697203242958446216014355797, 6.87439164003596768971370692637, 7.35405543094406606722650232925, 7.87952044240999256766427664650, 8.017079140951883488070335008157
