L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 11-s − 0.694·13-s − 14-s + 16-s + 2.69·17-s + 5.51·19-s + 20-s − 22-s + 7.43·23-s + 25-s + 0.694·26-s + 28-s + 0.610·29-s − 7.51·31-s − 32-s − 2.69·34-s + 35-s − 1.30·37-s − 5.51·38-s − 40-s + 9.51·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.301·11-s − 0.192·13-s − 0.267·14-s + 0.250·16-s + 0.653·17-s + 1.26·19-s + 0.223·20-s − 0.213·22-s + 1.55·23-s + 0.200·25-s + 0.136·26-s + 0.188·28-s + 0.113·29-s − 1.35·31-s − 0.176·32-s − 0.462·34-s + 0.169·35-s − 0.214·37-s − 0.895·38-s − 0.158·40-s + 1.48·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.923540698\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.923540698\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 0.694T + 13T^{2} \) |
| 17 | \( 1 - 2.69T + 17T^{2} \) |
| 19 | \( 1 - 5.51T + 19T^{2} \) |
| 23 | \( 1 - 7.43T + 23T^{2} \) |
| 29 | \( 1 - 0.610T + 29T^{2} \) |
| 31 | \( 1 + 7.51T + 31T^{2} \) |
| 37 | \( 1 + 1.30T + 37T^{2} \) |
| 41 | \( 1 - 9.51T + 41T^{2} \) |
| 43 | \( 1 - 5.51T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 4.12T + 53T^{2} \) |
| 59 | \( 1 + 3.51T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 8.95T + 67T^{2} \) |
| 71 | \( 1 - 4.77T + 71T^{2} \) |
| 73 | \( 1 - 5.51T + 73T^{2} \) |
| 79 | \( 1 + 2.77T + 79T^{2} \) |
| 83 | \( 1 + 4.73T + 83T^{2} \) |
| 89 | \( 1 - 8.04T + 89T^{2} \) |
| 97 | \( 1 + 6.90T + 97T^{2} \) |
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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
−7.83137692836985750234563017543, −7.39421500286946052716792106937, −6.77158758952053823195426772354, −5.75988713873773532899181444454, −5.36724751294884522783624321661, −4.40977347241790630803480861853, −3.34767078732276831058188571422, −2.65085672423893477198451842824, −1.58145068022609358610088140695, −0.851286187519780566946491845247,
0.851286187519780566946491845247, 1.58145068022609358610088140695, 2.65085672423893477198451842824, 3.34767078732276831058188571422, 4.40977347241790630803480861853, 5.36724751294884522783624321661, 5.75988713873773532899181444454, 6.77158758952053823195426772354, 7.39421500286946052716792106937, 7.83137692836985750234563017543