| L(s) = 1 | + 2·2-s + 3-s + 3·4-s + 2·6-s + 7-s + 4·8-s + 2·11-s + 3·12-s + 7·13-s + 2·14-s + 5·16-s + 8·17-s − 19-s + 21-s + 4·22-s − 8·23-s + 4·24-s + 5·25-s + 14·26-s − 27-s + 3·28-s − 6·29-s + 6·32-s + 2·33-s + 16·34-s − 22·37-s − 2·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.816·6-s + 0.377·7-s + 1.41·8-s + 0.603·11-s + 0.866·12-s + 1.94·13-s + 0.534·14-s + 5/4·16-s + 1.94·17-s − 0.229·19-s + 0.218·21-s + 0.852·22-s − 1.66·23-s + 0.816·24-s + 25-s + 2.74·26-s − 0.192·27-s + 0.566·28-s − 1.11·29-s + 1.06·32-s + 0.348·33-s + 2.74·34-s − 3.61·37-s − 0.324·38-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
| \(L(1)\) |
\(\approx\) |
\(6.510040320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.510040320\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 9 T - 16 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−11.09964038198210287488275915418, −10.65557465304804636025849951891, −10.29653702259233336677947772304, −9.891353316813217805081471538463, −9.118631635201404034200116848720, −8.762371160673402026823757619970, −8.146188579670760887720110155806, −8.077037503875349807103689417072, −7.17723107496324155920782566497, −7.02730290133114816217680512346, −6.18739781263418676226224057484, −5.95533321399453309972198892816, −5.42672055270669569752386182742, −5.02331455499802823959581565881, −4.06238778181118898769232757125, −3.87162468135973067170154787253, −3.35718472490131101407491853185, −2.91807774764654001259220595074, −1.63293253464463989546752092788, −1.57732999563791690131468045617,
1.57732999563791690131468045617, 1.63293253464463989546752092788, 2.91807774764654001259220595074, 3.35718472490131101407491853185, 3.87162468135973067170154787253, 4.06238778181118898769232757125, 5.02331455499802823959581565881, 5.42672055270669569752386182742, 5.95533321399453309972198892816, 6.18739781263418676226224057484, 7.02730290133114816217680512346, 7.17723107496324155920782566497, 8.077037503875349807103689417072, 8.146188579670760887720110155806, 8.762371160673402026823757619970, 9.118631635201404034200116848720, 9.891353316813217805081471538463, 10.29653702259233336677947772304, 10.65557465304804636025849951891, 11.09964038198210287488275915418
