| L(s) = 1 | + 2-s + 3·5-s − 8-s + 3·10-s − 3·11-s + 5·13-s − 16-s + 6·17-s − 10·19-s − 3·22-s − 3·23-s + 5·25-s + 5·26-s − 3·29-s − 4·31-s + 6·34-s − 14·37-s − 10·38-s − 3·40-s + 9·41-s − 11·43-s − 3·46-s + 5·50-s + 6·53-s − 9·55-s − 3·58-s − 12·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.34·5-s − 0.353·8-s + 0.948·10-s − 0.904·11-s + 1.38·13-s − 1/4·16-s + 1.45·17-s − 2.29·19-s − 0.639·22-s − 0.625·23-s + 25-s + 0.980·26-s − 0.557·29-s − 0.718·31-s + 1.02·34-s − 2.30·37-s − 1.62·38-s − 0.474·40-s + 1.40·41-s − 1.67·43-s − 0.442·46-s + 0.707·50-s + 0.824·53-s − 1.21·55-s − 0.393·58-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.189497559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.189497559\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 3 T - 74 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−9.276611754769035626486610107848, −8.692023977446466306932474226306, −8.356314692212496075984801633281, −8.028657432039688223380371144688, −7.36094156930555073315845791780, −7.21296147753085255839954426964, −6.48474734382392155601805669638, −6.14192590752460432442474261941, −5.99702657260126979482580883583, −5.65264997292962608590144504757, −5.06893992920965133905414139555, −4.99825373106986070631150151506, −4.17117241775305364188435689805, −3.97090046972027519131975859418, −3.25271442401967611321753458479, −3.14451635807868855489452187863, −2.32021936395067499654888973360, −1.77934422631694402929661057909, −1.64738960895474321671234553994, −0.48897360983927927255896905391,
0.48897360983927927255896905391, 1.64738960895474321671234553994, 1.77934422631694402929661057909, 2.32021936395067499654888973360, 3.14451635807868855489452187863, 3.25271442401967611321753458479, 3.97090046972027519131975859418, 4.17117241775305364188435689805, 4.99825373106986070631150151506, 5.06893992920965133905414139555, 5.65264997292962608590144504757, 5.99702657260126979482580883583, 6.14192590752460432442474261941, 6.48474734382392155601805669638, 7.21296147753085255839954426964, 7.36094156930555073315845791780, 8.028657432039688223380371144688, 8.356314692212496075984801633281, 8.692023977446466306932474226306, 9.276611754769035626486610107848
