| L(s) = 1 | − 4·3-s − 3·7-s + 10·9-s + 5·11-s − 4·13-s − 7·17-s + 3·19-s + 12·21-s − 8·23-s − 20·27-s + 2·29-s + 5·31-s − 20·33-s + 37-s + 16·39-s + 7·41-s − 6·43-s − 8·49-s + 28·51-s − 6·53-s − 12·57-s − 9·59-s + 19·61-s − 30·63-s + 4·67-s + 32·69-s + 19·71-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1.13·7-s + 10/3·9-s + 1.50·11-s − 1.10·13-s − 1.69·17-s + 0.688·19-s + 2.61·21-s − 1.66·23-s − 3.84·27-s + 0.371·29-s + 0.898·31-s − 3.48·33-s + 0.164·37-s + 2.56·39-s + 1.09·41-s − 0.914·43-s − 8/7·49-s + 3.92·51-s − 0.824·53-s − 1.58·57-s − 1.17·59-s + 2.43·61-s − 3.77·63-s + 0.488·67-s + 3.85·69-s + 2.25·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.008627701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.008627701\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $C_2 \wr S_4$ | \( 1 + 3 T + 17 T^{2} + 24 T^{3} + 120 T^{4} + 24 p T^{5} + 17 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 5 T + 15 T^{2} - 6 T^{3} + 16 T^{4} - 6 p T^{5} + 15 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 7 T + 39 T^{2} + 186 T^{3} + 964 T^{4} + 186 p T^{5} + 39 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 3 T + 34 T^{2} - 27 T^{3} + 606 T^{4} - 27 p T^{5} + 34 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 8 T + 84 T^{2} + 420 T^{3} + 2614 T^{4} + 420 p T^{5} + 84 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 2 T + 60 T^{2} + 36 T^{3} + 1741 T^{4} + 36 p T^{5} + 60 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 5 T + 79 T^{2} - 158 T^{3} + 2548 T^{4} - 158 p T^{5} + 79 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - T + 2 p T^{2} - 123 T^{3} + 3374 T^{4} - 123 p T^{5} + 2 p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 7 T + 72 T^{2} + 99 T^{3} + 298 T^{4} + 99 p T^{5} + 72 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 6 T + 80 T^{2} + 714 T^{3} + 3174 T^{4} + 714 p T^{5} + 80 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $S_4\times C_2$ | \( 1 + 60 T^{2} - 18 T^{3} + 5171 T^{4} - 18 p T^{5} + 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 6 T - 12 T^{2} - 24 T^{3} + 3257 T^{4} - 24 p T^{5} - 12 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 9 T + 107 T^{2} + 576 T^{3} + 6690 T^{4} + 576 p T^{5} + 107 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 19 T + 319 T^{2} - 3286 T^{3} + 30796 T^{4} - 3286 p T^{5} + 319 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 4 T + 142 T^{2} - 382 T^{3} + 12817 T^{4} - 382 p T^{5} + 142 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 19 T + 374 T^{2} - 3967 T^{3} + 42262 T^{4} - 3967 p T^{5} + 374 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 6 T + 16 T^{2} - 486 T^{3} + 11694 T^{4} - 486 p T^{5} + 16 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 9 T + 272 T^{2} + 1893 T^{3} + 30882 T^{4} + 1893 p T^{5} + 272 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 21 T + 483 T^{2} + 5676 T^{3} + 66866 T^{4} + 5676 p T^{5} + 483 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 14 T + 276 T^{2} - 3402 T^{3} + 34726 T^{4} - 3402 p T^{5} + 276 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 6 T + 344 T^{2} - 1434 T^{3} + 47598 T^{4} - 1434 p T^{5} + 344 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.67750434039793664449589351225, −5.19638382642675835643222051274, −5.09969265022228598563053428859, −5.07556610254713798432033766266, −4.90855440244577630247411660590, −4.58950223956366971765316841324, −4.31203436199420879128053259538, −4.25789914645874389154547687136, −4.21370361099743871366204623603, −3.92983209274810142688853627879, −3.68573879355457138829176699388, −3.46758675196988084927849899090, −3.42830709720399805649472889230, −2.90681412948234669084204512903, −2.70053576957027628798496210121, −2.63262926003324853705069689580, −2.45875949162226048348238878904, −1.86302681645402985529313277070, −1.72466624129393740727678591897, −1.66696896130371117897900726874, −1.55464762127245655000248891960, −0.813636955478855706489784502271, −0.65542749639826579741581481344, −0.57463280764360192965629254333, −0.23631501577451695489503474969,
0.23631501577451695489503474969, 0.57463280764360192965629254333, 0.65542749639826579741581481344, 0.813636955478855706489784502271, 1.55464762127245655000248891960, 1.66696896130371117897900726874, 1.72466624129393740727678591897, 1.86302681645402985529313277070, 2.45875949162226048348238878904, 2.63262926003324853705069689580, 2.70053576957027628798496210121, 2.90681412948234669084204512903, 3.42830709720399805649472889230, 3.46758675196988084927849899090, 3.68573879355457138829176699388, 3.92983209274810142688853627879, 4.21370361099743871366204623603, 4.25789914645874389154547687136, 4.31203436199420879128053259538, 4.58950223956366971765316841324, 4.90855440244577630247411660590, 5.07556610254713798432033766266, 5.09969265022228598563053428859, 5.19638382642675835643222051274, 5.67750434039793664449589351225
