| L(s) = 1 | + 2·3-s − 2·5-s − 7-s + 3·9-s − 4·11-s + 2·13-s − 4·15-s + 4·19-s − 2·21-s − 2·25-s + 4·27-s − 3·29-s − 19·31-s − 8·33-s + 2·35-s − 16·37-s + 4·39-s + 9·41-s − 6·45-s − 15·47-s − 12·49-s − 5·53-s + 8·55-s + 8·57-s + 7·59-s + 10·61-s − 3·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.377·7-s + 9-s − 1.20·11-s + 0.554·13-s − 1.03·15-s + 0.917·19-s − 0.436·21-s − 2/5·25-s + 0.769·27-s − 0.557·29-s − 3.41·31-s − 1.39·33-s + 0.338·35-s − 2.63·37-s + 0.640·39-s + 1.40·41-s − 0.894·45-s − 2.18·47-s − 1.71·49-s − 0.686·53-s + 1.07·55-s + 1.05·57-s + 0.911·59-s + 1.28·61-s − 0.377·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25522704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25522704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 421 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 59 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 19 T + 151 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 15 T + 149 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 111 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 7 T + 129 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 142 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 19 T + 193 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 13 T + 183 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 14 T + 190 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - T + 127 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9 T + 175 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 27 T + 345 T^{2} + 27 p T^{3} + p^{2} T^{4} \) |
| 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
−8.074868815740760521241104366755, −7.74352449392757349682853717269, −7.34602084665853768460558291203, −7.22549309020668630959924879901, −6.81049868604162186826559318459, −6.34675938817679873821521313628, −5.82222857628763429083069237453, −5.39629899229291060663021182825, −5.04883518287873073938759349116, −4.89537708164672777682480893805, −4.03756220821848827726373019309, −3.75632177918213904129192165190, −3.42842569397933790886637532429, −3.35663748696204764116191140646, −2.72338192403924548091642364767, −2.10322742605018277074030336953, −1.80871924620936632998895855387, −1.24302435981079440958072077896, 0, 0,
1.24302435981079440958072077896, 1.80871924620936632998895855387, 2.10322742605018277074030336953, 2.72338192403924548091642364767, 3.35663748696204764116191140646, 3.42842569397933790886637532429, 3.75632177918213904129192165190, 4.03756220821848827726373019309, 4.89537708164672777682480893805, 5.04883518287873073938759349116, 5.39629899229291060663021182825, 5.82222857628763429083069237453, 6.34675938817679873821521313628, 6.81049868604162186826559318459, 7.22549309020668630959924879901, 7.34602084665853768460558291203, 7.74352449392757349682853717269, 8.074868815740760521241104366755
