| L(s) = 1 | − 2·2-s + 3·4-s + 3·5-s − 7-s − 4·8-s − 6·10-s + 4·13-s + 2·14-s + 5·16-s + 6·17-s + 4·19-s + 9·20-s + 6·23-s + 5·25-s − 8·26-s − 3·28-s − 3·29-s + 16·31-s − 6·32-s − 12·34-s − 3·35-s − 8·37-s − 8·38-s − 12·40-s − 6·41-s − 8·43-s − 12·46-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 1.34·5-s − 0.377·7-s − 1.41·8-s − 1.89·10-s + 1.10·13-s + 0.534·14-s + 5/4·16-s + 1.45·17-s + 0.917·19-s + 2.01·20-s + 1.25·23-s + 25-s − 1.56·26-s − 0.566·28-s − 0.557·29-s + 2.87·31-s − 1.06·32-s − 2.05·34-s − 0.507·35-s − 1.31·37-s − 1.29·38-s − 1.89·40-s − 0.937·41-s − 1.21·43-s − 1.76·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.678637509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.678637509\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−10.14956331170890471153454533301, −9.563879823023061226078177454463, −9.280973068077343621633766763669, −8.909845541773668939683782046187, −8.424082234198822255006942348559, −8.124997421604800971702105607134, −7.67224223603255596966376839082, −7.16444692084898744562597660190, −6.58581498595970786976203595444, −6.41991919044478154507317534546, −6.05661535867833975740730199557, −5.31972660538423526136429841017, −5.22965819784247440777917897026, −4.48179559245567463677681177920, −3.32945796171510404594180785747, −3.24282323564344471485772370994, −2.79093233604965698282726990498, −1.69472685761918745251955579675, −1.51170455996311713644398502063, −0.77593272133235155404951997776,
0.77593272133235155404951997776, 1.51170455996311713644398502063, 1.69472685761918745251955579675, 2.79093233604965698282726990498, 3.24282323564344471485772370994, 3.32945796171510404594180785747, 4.48179559245567463677681177920, 5.22965819784247440777917897026, 5.31972660538423526136429841017, 6.05661535867833975740730199557, 6.41991919044478154507317534546, 6.58581498595970786976203595444, 7.16444692084898744562597660190, 7.67224223603255596966376839082, 8.124997421604800971702105607134, 8.424082234198822255006942348559, 8.909845541773668939683782046187, 9.280973068077343621633766763669, 9.563879823023061226078177454463, 10.14956331170890471153454533301
