L(s) = 1 | + 2-s − 3-s + 4-s + 3.34·5-s − 6-s + 7-s + 8-s + 9-s + 3.34·10-s + 11-s − 12-s − 13-s + 14-s − 3.34·15-s + 16-s + 1.19·17-s + 18-s − 5.08·19-s + 3.34·20-s − 21-s + 22-s + 5.00·23-s − 24-s + 6.20·25-s − 26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.49·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 1.05·10-s + 0.301·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.864·15-s + 0.250·16-s + 0.289·17-s + 0.235·18-s − 1.16·19-s + 0.748·20-s − 0.218·21-s + 0.213·22-s + 1.04·23-s − 0.204·24-s + 1.24·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.963385541\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.963385541\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 3.34T + 5T^{2} \) |
| 17 | \( 1 - 1.19T + 17T^{2} \) |
| 19 | \( 1 + 5.08T + 19T^{2} \) |
| 23 | \( 1 - 5.00T + 23T^{2} \) |
| 29 | \( 1 - 0.676T + 29T^{2} \) |
| 31 | \( 1 + 3.96T + 31T^{2} \) |
| 37 | \( 1 - 4.15T + 37T^{2} \) |
| 41 | \( 1 + 8.08T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 - 7.28T + 47T^{2} \) |
| 53 | \( 1 - 0.0518T + 53T^{2} \) |
| 59 | \( 1 - 1.14T + 59T^{2} \) |
| 61 | \( 1 - 1.74T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 + 5.54T + 71T^{2} \) |
| 73 | \( 1 - 6.13T + 73T^{2} \) |
| 79 | \( 1 - 5.47T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 + 17.9T + 89T^{2} \) |
| 97 | \( 1 + 1.53T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.958847675560838233023232884801, −7.00090465526876279770356185217, −6.54173033403341135499077722282, −5.77524699581616698927072383042, −5.35258921963159395605095443551, −4.63126093783216425318458766086, −3.81101299363727401489615069033, −2.61054475054743787446983244002, −1.98750577916672607857543203825, −1.02057578445559393145909080677,
1.02057578445559393145909080677, 1.98750577916672607857543203825, 2.61054475054743787446983244002, 3.81101299363727401489615069033, 4.63126093783216425318458766086, 5.35258921963159395605095443551, 5.77524699581616698927072383042, 6.54173033403341135499077722282, 7.00090465526876279770356185217, 7.958847675560838233023232884801