| L(s) = 1 | − 2·2-s − 13·4-s + 44·8-s − 10·9-s + 116·13-s + 101·16-s + 34·17-s + 20·18-s + 160·19-s − 232·26-s − 614·32-s − 68·34-s + 130·36-s − 320·38-s − 544·43-s + 928·47-s + 490·49-s − 1.50e3·52-s − 1.28e3·53-s − 360·59-s − 361·64-s + 1.84e3·67-s − 442·68-s − 440·72-s − 2.08e3·76-s − 629·81-s − 1.10e3·83-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.62·4-s + 1.94·8-s − 0.370·9-s + 2.47·13-s + 1.57·16-s + 0.485·17-s + 0.261·18-s + 1.93·19-s − 1.74·26-s − 3.39·32-s − 0.342·34-s + 0.601·36-s − 1.36·38-s − 1.92·43-s + 2.88·47-s + 10/7·49-s − 4.02·52-s − 3.32·53-s − 0.794·59-s − 0.705·64-s + 3.36·67-s − 0.788·68-s − 0.720·72-s − 3.13·76-s − 0.862·81-s − 1.45·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.532928674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.532928674\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 - 2 p T + p^{3} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p^{3} T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + 10 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2262 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 58 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 80 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10410 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 32902 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 54682 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 83350 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 127842 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 272 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 464 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 642 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 180 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 441862 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 924 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 707722 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 92450 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 793478 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 552 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 1490 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 68030 T^{2} + p^{6} T^{4} \) |
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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
−10.77031262346374251875306629917, −10.46008187906253977505131536574, −10.01324110323010177715470449745, −9.385482675197433646936123831792, −9.270748603190843204829023757068, −8.613925334578762722110399497321, −8.534656107555749445681663459615, −7.79175214717372570323127820882, −7.70225044363267558479617561804, −6.91758270861956414710879069314, −5.98410985668681452200137246172, −5.87156800846396030581935525975, −5.08810310363216996046627288173, −4.83174077862578454627207069071, −3.88328923137175042867369871647, −3.61503695518772964617453264286, −3.15175839465597723304440587570, −1.70350788806466617721777604432, −1.01349883565595495735483123897, −0.64409132693716841096213039130,
0.64409132693716841096213039130, 1.01349883565595495735483123897, 1.70350788806466617721777604432, 3.15175839465597723304440587570, 3.61503695518772964617453264286, 3.88328923137175042867369871647, 4.83174077862578454627207069071, 5.08810310363216996046627288173, 5.87156800846396030581935525975, 5.98410985668681452200137246172, 6.91758270861956414710879069314, 7.70225044363267558479617561804, 7.79175214717372570323127820882, 8.534656107555749445681663459615, 8.613925334578762722110399497321, 9.270748603190843204829023757068, 9.385482675197433646936123831792, 10.01324110323010177715470449745, 10.46008187906253977505131536574, 10.77031262346374251875306629917
