| L(s) = 1 | − 4-s + 2·5-s + 3·7-s + 2·9-s − 2·13-s + 16-s − 2·20-s + 2·25-s − 3·28-s − 8·29-s + 6·35-s − 2·36-s + 4·45-s + 6·49-s + 2·52-s + 4·53-s − 10·59-s + 6·63-s − 64-s − 4·65-s − 8·67-s + 8·71-s + 2·80-s − 5·81-s − 2·83-s − 6·91-s − 2·100-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.894·5-s + 1.13·7-s + 2/3·9-s − 0.554·13-s + 1/4·16-s − 0.447·20-s + 2/5·25-s − 0.566·28-s − 1.48·29-s + 1.01·35-s − 1/3·36-s + 0.596·45-s + 6/7·49-s + 0.277·52-s + 0.549·53-s − 1.30·59-s + 0.755·63-s − 1/8·64-s − 0.496·65-s − 0.977·67-s + 0.949·71-s + 0.223·80-s − 5/9·81-s − 0.219·83-s − 0.628·91-s − 1/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23548 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23548 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.396415711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.396415711\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
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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
−10.80404830881719475219388908999, −10.04652465842475049039963364295, −9.731710784914762882272254341544, −9.162112345516062564890988652675, −8.680373316617984887938249809235, −7.999925405017203766641651580562, −7.45074358256423247875113409431, −6.99961983467005104851457580996, −6.04485271199479270426447995259, −5.57646187604123617125678826396, −4.86845228270712524403722482663, −4.43933224082992873393526002196, −3.53356305219019962161480660547, −2.31723985951592697337671239841, −1.50633743693629636857802515312,
1.50633743693629636857802515312, 2.31723985951592697337671239841, 3.53356305219019962161480660547, 4.43933224082992873393526002196, 4.86845228270712524403722482663, 5.57646187604123617125678826396, 6.04485271199479270426447995259, 6.99961983467005104851457580996, 7.45074358256423247875113409431, 7.999925405017203766641651580562, 8.680373316617984887938249809235, 9.162112345516062564890988652675, 9.731710784914762882272254341544, 10.04652465842475049039963364295, 10.80404830881719475219388908999
