| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s + 2·7-s + 4·8-s − 4·10-s + 4·11-s + 8·13-s + 4·14-s + 5·16-s − 2·17-s − 2·19-s − 6·20-s + 8·22-s + 3·25-s + 16·26-s + 6·28-s − 4·29-s + 10·31-s + 6·32-s − 4·34-s − 4·35-s − 4·38-s − 8·40-s − 2·41-s + 4·43-s + 12·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.755·7-s + 1.41·8-s − 1.26·10-s + 1.20·11-s + 2.21·13-s + 1.06·14-s + 5/4·16-s − 0.485·17-s − 0.458·19-s − 1.34·20-s + 1.70·22-s + 3/5·25-s + 3.13·26-s + 1.13·28-s − 0.742·29-s + 1.79·31-s + 1.06·32-s − 0.685·34-s − 0.676·35-s − 0.648·38-s − 1.26·40-s − 0.312·41-s + 0.609·43-s + 1.80·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2924100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2924100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.923561293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.923561293\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 66 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 166 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 22 T + 270 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−9.377066340388711656351793919491, −9.029892591486118806802293488944, −8.505170069146041669368657992845, −8.473886372567864750953859992651, −7.76917616932480317152487608962, −7.64656640993540486062457393481, −6.88999181913416250631964355816, −6.63283353310251304038819269910, −6.23941617932906518089671523408, −6.02648691879393082886838566691, −5.29048096807453615851225433327, −5.03177567486546073289714433531, −4.37574047242206370391567980316, −4.05195296127906493418242332683, −3.68938720040460025268646006062, −3.59658785251440597450313424611, −2.61640077576820863844940973511, −2.23116079417947937357191947687, −1.30381797599188570940269305058, −1.00477982265802441032029886721,
1.00477982265802441032029886721, 1.30381797599188570940269305058, 2.23116079417947937357191947687, 2.61640077576820863844940973511, 3.59658785251440597450313424611, 3.68938720040460025268646006062, 4.05195296127906493418242332683, 4.37574047242206370391567980316, 5.03177567486546073289714433531, 5.29048096807453615851225433327, 6.02648691879393082886838566691, 6.23941617932906518089671523408, 6.63283353310251304038819269910, 6.88999181913416250631964355816, 7.64656640993540486062457393481, 7.76917616932480317152487608962, 8.473886372567864750953859992651, 8.505170069146041669368657992845, 9.029892591486118806802293488944, 9.377066340388711656351793919491
