L(s) = 1 | + 2-s + 3-s + 4-s + 1.35·5-s + 6-s − 7-s + 8-s + 9-s + 1.35·10-s + 1.30·11-s + 12-s − 13-s − 14-s + 1.35·15-s + 16-s + 17-s + 18-s + 3.46·19-s + 1.35·20-s − 21-s + 1.30·22-s − 8.03·23-s + 24-s − 3.16·25-s − 26-s + 27-s − 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.605·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.428·10-s + 0.392·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.349·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 0.795·19-s + 0.302·20-s − 0.218·21-s + 0.277·22-s − 1.67·23-s + 0.204·24-s − 0.633·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.871277927\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.871277927\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - 1.35T + 5T^{2} \) |
| 11 | \( 1 - 1.30T + 11T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 + 8.03T + 23T^{2} \) |
| 29 | \( 1 - 3.71T + 29T^{2} \) |
| 31 | \( 1 + 2.76T + 31T^{2} \) |
| 37 | \( 1 - 3.21T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 8.56T + 43T^{2} \) |
| 47 | \( 1 + 9.70T + 47T^{2} \) |
| 53 | \( 1 - 7.96T + 53T^{2} \) |
| 59 | \( 1 - 6.31T + 59T^{2} \) |
| 61 | \( 1 - 5.28T + 61T^{2} \) |
| 67 | \( 1 - 8.54T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 7.72T + 73T^{2} \) |
| 79 | \( 1 - 5.39T + 79T^{2} \) |
| 83 | \( 1 - 6.44T + 83T^{2} \) |
| 89 | \( 1 + 3.02T + 89T^{2} \) |
| 97 | \( 1 - 7.75T + 97T^{2} \) |
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
−7.72002775458930353189518046481, −6.93915825931405377320575989437, −6.22593599582166994748515115339, −5.71911717372018513203663867324, −4.96270489886685255998409387930, −3.98926476987821831752185228529, −3.62534512947718367313540910720, −2.53921535408047024354519408256, −2.10508759863490515173153212561, −0.938432960001158642413707331612,
0.938432960001158642413707331612, 2.10508759863490515173153212561, 2.53921535408047024354519408256, 3.62534512947718367313540910720, 3.98926476987821831752185228529, 4.96270489886685255998409387930, 5.71911717372018513203663867324, 6.22593599582166994748515115339, 6.93915825931405377320575989437, 7.72002775458930353189518046481