| L(s) = 1 | − 2-s − 6·3-s + 8·4-s + 5·5-s + 6·6-s − 23·8-s + 27·9-s − 5·10-s + 44·11-s − 48·12-s + 12·13-s − 30·15-s + 23·16-s + 24·17-s − 27·18-s + 114·19-s + 40·20-s − 44·22-s + 52·23-s + 138·24-s − 12·26-s − 270·27-s + 292·29-s + 30·30-s + 276·31-s − 184·32-s − 264·33-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 1.15·3-s + 4-s + 0.447·5-s + 0.408·6-s − 1.01·8-s + 9-s − 0.158·10-s + 1.20·11-s − 1.15·12-s + 0.256·13-s − 0.516·15-s + 0.359·16-s + 0.342·17-s − 0.353·18-s + 1.37·19-s + 0.447·20-s − 0.426·22-s + 0.471·23-s + 1.17·24-s − 0.0905·26-s − 1.92·27-s + 1.86·29-s + 0.182·30-s + 1.59·31-s − 1.01·32-s − 1.39·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.546370350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.546370350\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T - 7 T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 p T + p^{2} T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 p T + 5 p^{2} T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 24 T - 4337 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 p T + 17 p^{2} T^{2} - 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 52 T - 9463 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 146 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 276 T + 46385 T^{2} - 276 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 210 T - 6553 T^{2} - 210 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 444 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 492 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 612 T + 270721 T^{2} - 612 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 50 T - 146377 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 294 T - 118943 T^{2} + 294 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 450 T - 24481 T^{2} + 450 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 668 T + 145461 T^{2} - 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 308 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 12 T - 388873 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 596 T - 137823 T^{2} + 596 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 966 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 408 T - 538505 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1200 T + p^{3} T^{2} )^{2} \) |
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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
−12.00102756565324964025410279895, −11.25678430802501731174584087826, −10.93619716283625387006807379882, −10.82203710240911818944792014215, −9.846238316699220856678441707333, −9.489859009329122787085676245877, −9.304545361234816784745037962224, −8.553318273303221990727451696859, −7.71384568348536496250076251508, −7.30347559275397965106523910504, −6.88046747784604251184538767295, −6.13459079773251366318459466390, −5.81371951804520094728812141113, −5.72837591877963039331880292037, −4.29250986146726991854724055949, −4.28548507728475479085008487113, −2.87594876931707457599283412885, −2.52108905856689835759061612281, −1.03711906187881235699135225845, −1.00770248200830922281243532256,
1.00770248200830922281243532256, 1.03711906187881235699135225845, 2.52108905856689835759061612281, 2.87594876931707457599283412885, 4.28548507728475479085008487113, 4.29250986146726991854724055949, 5.72837591877963039331880292037, 5.81371951804520094728812141113, 6.13459079773251366318459466390, 6.88046747784604251184538767295, 7.30347559275397965106523910504, 7.71384568348536496250076251508, 8.553318273303221990727451696859, 9.304545361234816784745037962224, 9.489859009329122787085676245877, 9.846238316699220856678441707333, 10.82203710240911818944792014215, 10.93619716283625387006807379882, 11.25678430802501731174584087826, 12.00102756565324964025410279895
