| L(s) = 1 | + 0.197·2-s − 1.56·3-s − 1.96·4-s + 5-s − 0.308·6-s − 0.780·8-s − 0.544·9-s + 0.197·10-s + 11-s + 3.07·12-s + 0.133·13-s − 1.56·15-s + 3.76·16-s + 3.07·17-s − 0.107·18-s − 6.46·19-s − 1.96·20-s + 0.197·22-s + 4.38·23-s + 1.22·24-s + 25-s + 0.0263·26-s + 5.55·27-s − 8.40·29-s − 0.308·30-s + 0.406·31-s + 2.30·32-s + ⋯ |
| L(s) = 1 | + 0.139·2-s − 0.904·3-s − 0.980·4-s + 0.447·5-s − 0.126·6-s − 0.276·8-s − 0.181·9-s + 0.0623·10-s + 0.301·11-s + 0.887·12-s + 0.0371·13-s − 0.404·15-s + 0.942·16-s + 0.745·17-s − 0.0253·18-s − 1.48·19-s − 0.438·20-s + 0.0420·22-s + 0.913·23-s + 0.249·24-s + 0.200·25-s + 0.00517·26-s + 1.06·27-s − 1.56·29-s − 0.0563·30-s + 0.0730·31-s + 0.407·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 0.197T + 2T^{2} \) |
| 3 | \( 1 + 1.56T + 3T^{2} \) |
| 13 | \( 1 - 0.133T + 13T^{2} \) |
| 17 | \( 1 - 3.07T + 17T^{2} \) |
| 19 | \( 1 + 6.46T + 19T^{2} \) |
| 23 | \( 1 - 4.38T + 23T^{2} \) |
| 29 | \( 1 + 8.40T + 29T^{2} \) |
| 31 | \( 1 - 0.406T + 31T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 - 9.66T + 41T^{2} \) |
| 43 | \( 1 - 5.18T + 43T^{2} \) |
| 47 | \( 1 - 1.83T + 47T^{2} \) |
| 53 | \( 1 + 5.21T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 1.37T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 + 6.88T + 71T^{2} \) |
| 73 | \( 1 + 0.396T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 9.46T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−8.693396400768613526194279268550, −7.71265121565295896009625368341, −6.74651818791815729929030114319, −5.81955142566184477737731738357, −5.54565342043836646995231996469, −4.59293985587225110434812677176, −3.85728002479336887174860874670, −2.70039455305446347188923667464, −1.23706042285563580840895093929, 0,
1.23706042285563580840895093929, 2.70039455305446347188923667464, 3.85728002479336887174860874670, 4.59293985587225110434812677176, 5.54565342043836646995231996469, 5.81955142566184477737731738357, 6.74651818791815729929030114319, 7.71265121565295896009625368341, 8.693396400768613526194279268550
