| L(s) = 1 | + 3·3-s − 2·5-s + 2·9-s + 4·13-s − 6·15-s + 17-s − 19-s + 12·23-s + 3·25-s − 6·27-s + 14·29-s − 6·31-s − 4·37-s + 12·39-s − 3·41-s − 43-s − 4·45-s + 18·47-s − 14·49-s + 3·51-s − 16·53-s − 3·57-s − 3·59-s − 2·61-s − 8·65-s − 67-s + 36·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.894·5-s + 2/3·9-s + 1.10·13-s − 1.54·15-s + 0.242·17-s − 0.229·19-s + 2.50·23-s + 3/5·25-s − 1.15·27-s + 2.59·29-s − 1.07·31-s − 0.657·37-s + 1.92·39-s − 0.468·41-s − 0.152·43-s − 0.596·45-s + 2.62·47-s − 2·49-s + 0.420·51-s − 2.19·53-s − 0.397·57-s − 0.390·59-s − 0.256·61-s − 0.992·65-s − 0.122·67-s + 4.33·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.340381274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.340381274\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 27 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 14 T + 102 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 73 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T + 75 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T + 78 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 133 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 14 T + 186 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 155 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 178 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 27 T + 347 T^{2} - 27 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 149 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 13 T + 235 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−7.898360305396651243282540210048, −7.81171298440590138550951882914, −7.23652973786037410450576810856, −6.82140508404195405100321489740, −6.61649924580694762656716892089, −6.40736916510283592252802729858, −5.76019710483915196380563740657, −5.40030201145246263948358393666, −4.91872551943209543526570266093, −4.78892530165371127776240215527, −4.19850692915880682827366725828, −3.90775135715675256084181539903, −3.33860949763299338313326608701, −3.27030707995317424181345704466, −2.85308006667421487722597513257, −2.77503431446389412983012847998, −1.87400985500840498422032604853, −1.72388669124427853003844161559, −0.870867948564871638575924756506, −0.61543355811441385667490015806,
0.61543355811441385667490015806, 0.870867948564871638575924756506, 1.72388669124427853003844161559, 1.87400985500840498422032604853, 2.77503431446389412983012847998, 2.85308006667421487722597513257, 3.27030707995317424181345704466, 3.33860949763299338313326608701, 3.90775135715675256084181539903, 4.19850692915880682827366725828, 4.78892530165371127776240215527, 4.91872551943209543526570266093, 5.40030201145246263948358393666, 5.76019710483915196380563740657, 6.40736916510283592252802729858, 6.61649924580694762656716892089, 6.82140508404195405100321489740, 7.23652973786037410450576810856, 7.81171298440590138550951882914, 7.898360305396651243282540210048
