| L(s) = 1 | − 2-s − 4·5-s − 3·7-s + 8-s + 4·10-s − 4·11-s + 3·13-s + 3·14-s − 16-s − 6·17-s + 4·19-s + 4·22-s − 12·23-s + 5·25-s − 3·26-s + 14·31-s + 6·34-s + 12·35-s − 10·37-s − 4·38-s − 4·40-s − 6·41-s + 2·43-s + 12·46-s + 20·47-s + 7·49-s − 5·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.78·5-s − 1.13·7-s + 0.353·8-s + 1.26·10-s − 1.20·11-s + 0.832·13-s + 0.801·14-s − 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.852·22-s − 2.50·23-s + 25-s − 0.588·26-s + 2.51·31-s + 1.02·34-s + 2.02·35-s − 1.64·37-s − 0.648·38-s − 0.632·40-s − 0.937·41-s + 0.304·43-s + 1.76·46-s + 2.91·47-s + 49-s − 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 443556 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 37 | $C_2$ | \( 1 + 10 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 9 T + 14 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 9 T + 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 10 T + 17 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
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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
−10.31286164357834791772723009855, −9.966659234082782615073459804630, −9.406007182303430414218172888256, −8.908895212633564968781330639104, −8.407816401649645240515948373755, −8.282178749709244479117173077931, −7.63114409069224186777353796882, −7.54700846199185879743343822324, −6.75482308557453052827244452093, −6.56484979197247786049064135226, −5.67803062864176670050540150059, −5.53860173395444301869005545482, −4.33877977110194943733969098680, −4.22452839744698285560234779314, −3.87051719814362659591798645681, −2.86448527747365430718769218154, −2.72765823607147602467199813816, −1.44498062970319745650431633197, 0, 0,
1.44498062970319745650431633197, 2.72765823607147602467199813816, 2.86448527747365430718769218154, 3.87051719814362659591798645681, 4.22452839744698285560234779314, 4.33877977110194943733969098680, 5.53860173395444301869005545482, 5.67803062864176670050540150059, 6.56484979197247786049064135226, 6.75482308557453052827244452093, 7.54700846199185879743343822324, 7.63114409069224186777353796882, 8.282178749709244479117173077931, 8.407816401649645240515948373755, 8.908895212633564968781330639104, 9.406007182303430414218172888256, 9.966659234082782615073459804630, 10.31286164357834791772723009855
