L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 5·11-s − 12-s − 5·13-s − 14-s + 15-s + 16-s + 2·17-s − 18-s + 3·19-s − 20-s − 21-s − 5·22-s + 23-s + 24-s + 25-s + 5·26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.50·11-s − 0.288·12-s − 1.38·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.688·19-s − 0.223·20-s − 0.218·21-s − 1.06·22-s + 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.980·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.829269444\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.829269444\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 37 | \( 1 \) |
good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.35267087592924, −12.15654099295110, −11.80571512201753, −11.41533758200203, −10.95747625170542, −10.32586092445856, −10.00604955889623, −9.521054186115423, −9.064503269619738, −8.642084384011922, −8.020191742567574, −7.544305171503394, −7.164539829322878, −6.732421263453372, −6.295229827507744, −5.564060424131535, −5.207067749965090, −4.527153357953775, −4.186069762294530, −3.430939107938076, −2.967883493961256, −2.176807255072373, −1.621371643316441, −0.9097245617166723, −0.5412294567317200,
0.5412294567317200, 0.9097245617166723, 1.621371643316441, 2.176807255072373, 2.967883493961256, 3.430939107938076, 4.186069762294530, 4.527153357953775, 5.207067749965090, 5.564060424131535, 6.295229827507744, 6.732421263453372, 7.164539829322878, 7.544305171503394, 8.020191742567574, 8.642084384011922, 9.064503269619738, 9.521054186115423, 10.00604955889623, 10.32586092445856, 10.95747625170542, 11.41533758200203, 11.80571512201753, 12.15654099295110, 12.35267087592924