L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 1.56·7-s − 8-s + 9-s + 10-s + 11-s − 12-s − 5.12·13-s − 1.56·14-s + 15-s + 16-s − 17-s − 18-s − 4·19-s − 20-s − 1.56·21-s − 22-s + 2.43·23-s + 24-s + 25-s + 5.12·26-s − 27-s + 1.56·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 0.590·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s − 1.42·13-s − 0.417·14-s + 0.258·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.340·21-s − 0.213·22-s + 0.508·23-s + 0.204·24-s + 0.200·25-s + 1.00·26-s − 0.192·27-s + 0.295·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7136397671\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7136397671\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 - 1.56T + 7T^{2} \) |
| 13 | \( 1 + 5.12T + 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 + 0.438T + 29T^{2} \) |
| 31 | \( 1 + 8.68T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 1.12T + 41T^{2} \) |
| 43 | \( 1 - 7.80T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 2.87T + 53T^{2} \) |
| 59 | \( 1 - 3.12T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 4.87T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + 2.87T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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
−8.026177963500719243738330912203, −7.48742552501311536379806758440, −6.88117386658057782635738793468, −6.14561448837823693786039555317, −5.19602631471288819476757639719, −4.61449278280852034215899450891, −3.74034202015720985244498576941, −2.55063128601546482485882709861, −1.73973230800917795382689808226, −0.50995253548783907239935113684,
0.50995253548783907239935113684, 1.73973230800917795382689808226, 2.55063128601546482485882709861, 3.74034202015720985244498576941, 4.61449278280852034215899450891, 5.19602631471288819476757639719, 6.14561448837823693786039555317, 6.88117386658057782635738793468, 7.48742552501311536379806758440, 8.026177963500719243738330912203