| L(s) = 1 | + 2·2-s − 3·3-s + 3·4-s − 3·5-s − 6·6-s − 4·7-s + 4·8-s + 6·9-s − 6·10-s − 3·11-s − 9·12-s − 2·13-s − 8·14-s + 9·15-s + 5·16-s − 12·17-s + 12·18-s − 7·19-s − 9·20-s + 12·21-s − 6·22-s − 12·24-s + 25-s − 4·26-s − 9·27-s − 12·28-s − 15·29-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.73·3-s + 3/2·4-s − 1.34·5-s − 2.44·6-s − 1.51·7-s + 1.41·8-s + 2·9-s − 1.89·10-s − 0.904·11-s − 2.59·12-s − 0.554·13-s − 2.13·14-s + 2.32·15-s + 5/4·16-s − 2.91·17-s + 2.82·18-s − 1.60·19-s − 2.01·20-s + 2.61·21-s − 1.27·22-s − 2.44·24-s + 1/5·25-s − 0.784·26-s − 1.73·27-s − 2.26·28-s − 2.78·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 15 T + 104 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 3 T + 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T + 56 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 3 T + 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 9 T + 10 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.85191627918989288478667570194, −10.70494463142974212689201341470, −9.692079327670945531849115663895, −9.623667731938009357987880729243, −8.791601935706391686516811765657, −8.199739384402223562946311990849, −7.40374601234642795659912852462, −7.29223650595092413099063058109, −6.58685445446378410407689650865, −6.53730082681663336197910266044, −5.88217687117670662874406692742, −5.54096953136433769566023621046, −4.77636252571321673061547981874, −4.47408498124981164232762305979, −3.88710120605605338731745001363, −3.73809015703496643334743848650, −2.52524657819954168303774681925, −2.15835259747463672328322134200, 0, 0,
2.15835259747463672328322134200, 2.52524657819954168303774681925, 3.73809015703496643334743848650, 3.88710120605605338731745001363, 4.47408498124981164232762305979, 4.77636252571321673061547981874, 5.54096953136433769566023621046, 5.88217687117670662874406692742, 6.53730082681663336197910266044, 6.58685445446378410407689650865, 7.29223650595092413099063058109, 7.40374601234642795659912852462, 8.199739384402223562946311990849, 8.791601935706391686516811765657, 9.623667731938009357987880729243, 9.692079327670945531849115663895, 10.70494463142974212689201341470, 10.85191627918989288478667570194
