| L(s) = 1 | + 3-s + 2·5-s − 2·7-s − 2·9-s + 2·11-s − 2·13-s + 2·15-s + 9·17-s + 9·19-s − 2·21-s + 16·23-s + 3·25-s − 2·27-s − 7·29-s + 5·31-s + 2·33-s − 4·35-s + 3·37-s − 2·39-s − 5·41-s + 4·43-s − 4·45-s + 2·47-s + 3·49-s + 9·51-s + 2·53-s + 4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.755·7-s − 2/3·9-s + 0.603·11-s − 0.554·13-s + 0.516·15-s + 2.18·17-s + 2.06·19-s − 0.436·21-s + 3.33·23-s + 3/5·25-s − 0.384·27-s − 1.29·29-s + 0.898·31-s + 0.348·33-s − 0.676·35-s + 0.493·37-s − 0.320·39-s − 0.780·41-s + 0.609·43-s − 0.596·45-s + 0.291·47-s + 3/7·49-s + 1.26·51-s + 0.274·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52998400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52998400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.727789358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.727789358\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 9 T + 3 p T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 9 T + 55 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 41 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 5 T + 65 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T + 47 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 85 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 94 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T - 39 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 7 T + 117 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 130 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 158 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 11 T + 107 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 50 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 11 T + 205 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 178 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.940592565118556929540042495225, −7.75268517074002835564438271008, −7.36297738989938860766192743960, −7.06781397713876378009271005836, −6.75375457468286213414776593508, −6.47767649823608118735308305154, −5.74427833782162425349133540518, −5.62333922567089466448710665280, −5.29431606610796381835685171244, −5.24016072055114202110069411429, −4.52187750283015229486368213783, −4.13411806619104256058926754558, −3.30606630275508990391687579942, −3.30440747840295129030994366927, −3.00389795162504391202808891208, −2.76352627737758405324713883851, −2.09382477850061284249455356645, −1.47468673524018959690903899087, −0.885987301316753567224652147534, −0.809390120951690555309279674606,
0.809390120951690555309279674606, 0.885987301316753567224652147534, 1.47468673524018959690903899087, 2.09382477850061284249455356645, 2.76352627737758405324713883851, 3.00389795162504391202808891208, 3.30440747840295129030994366927, 3.30606630275508990391687579942, 4.13411806619104256058926754558, 4.52187750283015229486368213783, 5.24016072055114202110069411429, 5.29431606610796381835685171244, 5.62333922567089466448710665280, 5.74427833782162425349133540518, 6.47767649823608118735308305154, 6.75375457468286213414776593508, 7.06781397713876378009271005836, 7.36297738989938860766192743960, 7.75268517074002835564438271008, 7.940592565118556929540042495225
