L(s) = 1 | + 2·2-s + 3·4-s − 5-s − 5·7-s + 4·8-s − 2·10-s − 5·11-s − 10·14-s + 5·16-s + 4·17-s − 8·19-s − 3·20-s − 10·22-s + 4·23-s + 5·25-s − 15·28-s + 5·29-s + 6·31-s + 6·32-s + 8·34-s + 5·35-s + 4·37-s − 16·38-s − 4·40-s − 2·43-s − 15·44-s + 8·46-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.447·5-s − 1.88·7-s + 1.41·8-s − 0.632·10-s − 1.50·11-s − 2.67·14-s + 5/4·16-s + 0.970·17-s − 1.83·19-s − 0.670·20-s − 2.13·22-s + 0.834·23-s + 25-s − 2.83·28-s + 0.928·29-s + 1.07·31-s + 1.06·32-s + 1.37·34-s + 0.845·35-s + 0.657·37-s − 2.59·38-s − 0.632·40-s − 0.304·43-s − 2.26·44-s + 1.17·46-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
\(L(1)\) |
\(\approx\) |
\(2.599010994\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.599010994\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 7 T - 34 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
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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
−10.24325568620545621264617678335, −9.722815032513935836087037178044, −9.290587224480628138235668515716, −8.748594456877857504775649499084, −8.246954869797783614569292344123, −7.82696881808736457327883071137, −7.50963196714324918913973012470, −6.82963748780639103006076793117, −6.50085396884058343081308401086, −6.35268468604198654364906778511, −5.78913350808345790580688620004, −5.27967609763227623772526903188, −4.81077483172443124844892896363, −4.37094120031666981405274753454, −3.94943249384637026847873623693, −3.08468524151635009491501527226, −2.93240083776323606761493869618, −2.79876466069493092936522099232, −1.73792528971302509004567208480, −0.53574952050694922372747049499,
0.53574952050694922372747049499, 1.73792528971302509004567208480, 2.79876466069493092936522099232, 2.93240083776323606761493869618, 3.08468524151635009491501527226, 3.94943249384637026847873623693, 4.37094120031666981405274753454, 4.81077483172443124844892896363, 5.27967609763227623772526903188, 5.78913350808345790580688620004, 6.35268468604198654364906778511, 6.50085396884058343081308401086, 6.82963748780639103006076793117, 7.50963196714324918913973012470, 7.82696881808736457327883071137, 8.246954869797783614569292344123, 8.748594456877857504775649499084, 9.290587224480628138235668515716, 9.722815032513935836087037178044, 10.24325568620545621264617678335