| L(s) = 1 | + 2·3-s − 4·5-s + 4·7-s + 3·9-s − 2·11-s + 2·13-s − 8·15-s − 8·17-s − 4·19-s + 8·21-s − 4·23-s + 8·25-s + 4·27-s + 8·29-s − 8·31-s − 4·33-s − 16·35-s + 4·37-s + 4·39-s − 8·41-s − 4·43-s − 12·45-s − 2·49-s − 16·51-s − 12·53-s + 8·55-s − 8·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s + 1.51·7-s + 9-s − 0.603·11-s + 0.554·13-s − 2.06·15-s − 1.94·17-s − 0.917·19-s + 1.74·21-s − 0.834·23-s + 8/5·25-s + 0.769·27-s + 1.48·29-s − 1.43·31-s − 0.696·33-s − 2.70·35-s + 0.657·37-s + 0.640·39-s − 1.24·41-s − 0.609·43-s − 1.78·45-s − 2/7·49-s − 2.24·51-s − 1.64·53-s + 1.07·55-s − 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47114496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47114496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 44 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 68 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 74 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 84 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 144 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 138 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 108 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 176 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.919999109334025606221273760839, −7.76319348219339926835955019867, −7.07610748656616475632188864071, −6.93910561814564345404143395612, −6.50409544275740423117559880483, −6.28982308354127300045166465114, −5.49236546260907539230826188780, −5.14829275446428251582299742253, −4.67303633792060370723612930856, −4.47974541509801766756589621559, −4.21559310691597746324853040081, −3.83640295356849839790425653161, −3.41034442425935944666005628205, −3.05223049983477718929779064089, −2.40436029609541109796729580099, −2.17892950594738816622263988567, −1.61113803699920965864417313646, −1.21968883387872240987979421825, 0, 0,
1.21968883387872240987979421825, 1.61113803699920965864417313646, 2.17892950594738816622263988567, 2.40436029609541109796729580099, 3.05223049983477718929779064089, 3.41034442425935944666005628205, 3.83640295356849839790425653161, 4.21559310691597746324853040081, 4.47974541509801766756589621559, 4.67303633792060370723612930856, 5.14829275446428251582299742253, 5.49236546260907539230826188780, 6.28982308354127300045166465114, 6.50409544275740423117559880483, 6.93910561814564345404143395612, 7.07610748656616475632188864071, 7.76319348219339926835955019867, 7.919999109334025606221273760839
