| L(s) = 1 | + 2·3-s + 2·5-s − 2·7-s − 4·11-s + 10·13-s + 4·15-s + 4·17-s − 6·19-s − 4·21-s − 4·25-s − 2·27-s + 4·29-s + 8·31-s − 8·33-s − 4·35-s + 4·37-s + 20·39-s + 4·41-s − 4·43-s − 8·47-s + 3·49-s + 8·51-s + 24·53-s − 8·55-s − 12·57-s + 2·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.755·7-s − 1.20·11-s + 2.77·13-s + 1.03·15-s + 0.970·17-s − 1.37·19-s − 0.872·21-s − 4/5·25-s − 0.384·27-s + 0.742·29-s + 1.43·31-s − 1.39·33-s − 0.676·35-s + 0.657·37-s + 3.20·39-s + 0.624·41-s − 0.609·43-s − 1.16·47-s + 3/7·49-s + 1.12·51-s + 3.29·53-s − 1.07·55-s − 1.58·57-s + 0.260·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3211264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.188706359\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.188706359\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 10 T + 48 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 44 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 44 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T - 24 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.253040624711519209680139610122, −9.077653491582659354243822839238, −8.615530507070655399749285933414, −8.360032357897011971093428836927, −7.946863735446870377533255814020, −7.942631667902056021715410311813, −7.03322669551507082970663008197, −6.52661247881130734402729411472, −6.27520771628208186617024231429, −6.01290217464599460849970735609, −5.35585461296207624338836860423, −5.32452588655350428160513412822, −4.21857438777618662389274167994, −4.02280686643835579567134157531, −3.46869413088191558390132097313, −3.07206810596418610267219588385, −2.47151220982796584080128942268, −2.30081170083472991406564070794, −1.41069666208357436068907091684, −0.73758995759133091522638295855,
0.73758995759133091522638295855, 1.41069666208357436068907091684, 2.30081170083472991406564070794, 2.47151220982796584080128942268, 3.07206810596418610267219588385, 3.46869413088191558390132097313, 4.02280686643835579567134157531, 4.21857438777618662389274167994, 5.32452588655350428160513412822, 5.35585461296207624338836860423, 6.01290217464599460849970735609, 6.27520771628208186617024231429, 6.52661247881130734402729411472, 7.03322669551507082970663008197, 7.942631667902056021715410311813, 7.946863735446870377533255814020, 8.360032357897011971093428836927, 8.615530507070655399749285933414, 9.077653491582659354243822839238, 9.253040624711519209680139610122
