| L(s) = 1 | − 2-s − 3.41·3-s + 4-s + 1.61·5-s + 3.41·6-s − 1.52·7-s − 8-s + 8.67·9-s − 1.61·10-s − 1.89·11-s − 3.41·12-s − 3.18·13-s + 1.52·14-s − 5.52·15-s + 16-s + 1.69·17-s − 8.67·18-s + 3.67·19-s + 1.61·20-s + 5.19·21-s + 1.89·22-s − 2.69·23-s + 3.41·24-s − 2.39·25-s + 3.18·26-s − 19.4·27-s − 1.52·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.97·3-s + 0.5·4-s + 0.722·5-s + 1.39·6-s − 0.574·7-s − 0.353·8-s + 2.89·9-s − 0.510·10-s − 0.571·11-s − 0.986·12-s − 0.883·13-s + 0.406·14-s − 1.42·15-s + 0.250·16-s + 0.412·17-s − 2.04·18-s + 0.844·19-s + 0.361·20-s + 1.13·21-s + 0.404·22-s − 0.561·23-s + 0.697·24-s − 0.478·25-s + 0.624·26-s − 3.73·27-s − 0.287·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 3.41T + 3T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 7 | \( 1 + 1.52T + 7T^{2} \) |
| 11 | \( 1 + 1.89T + 11T^{2} \) |
| 13 | \( 1 + 3.18T + 13T^{2} \) |
| 17 | \( 1 - 1.69T + 17T^{2} \) |
| 19 | \( 1 - 3.67T + 19T^{2} \) |
| 23 | \( 1 + 2.69T + 23T^{2} \) |
| 31 | \( 1 - 1.78T + 31T^{2} \) |
| 37 | \( 1 - 7.07T + 37T^{2} \) |
| 41 | \( 1 - 5.03T + 41T^{2} \) |
| 43 | \( 1 - 1.89T + 43T^{2} \) |
| 47 | \( 1 + 2.57T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 - 2.95T + 59T^{2} \) |
| 61 | \( 1 - 7.74T + 61T^{2} \) |
| 67 | \( 1 - 2.47T + 67T^{2} \) |
| 71 | \( 1 - 6.06T + 71T^{2} \) |
| 73 | \( 1 + 3.74T + 73T^{2} \) |
| 79 | \( 1 + 9.22T + 79T^{2} \) |
| 83 | \( 1 + 5.52T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−9.491280252919751660318713113705, −7.923906410911647185758370628821, −7.20761301954186033087634011418, −6.47697304193846975504108272103, −5.70829067831697462114606752740, −5.28056377802281550576302211956, −4.12394005569430865376691996099, −2.50357072987484932210345331178, −1.19884447357088848587887105758, 0,
1.19884447357088848587887105758, 2.50357072987484932210345331178, 4.12394005569430865376691996099, 5.28056377802281550576302211956, 5.70829067831697462114606752740, 6.47697304193846975504108272103, 7.20761301954186033087634011418, 7.923906410911647185758370628821, 9.491280252919751660318713113705
