| L(s) = 1 | − 2.23·2-s − 1.61·3-s + 3.00·4-s − 5-s + 3.61·6-s + 7-s − 2.23·8-s − 0.381·9-s + 2.23·10-s − 4.85·12-s − 2.38·13-s − 2.23·14-s + 1.61·15-s − 0.999·16-s − 3.38·17-s + 0.854·18-s − 1.23·19-s − 3.00·20-s − 1.61·21-s + 4.47·23-s + 3.61·24-s + 25-s + 5.32·26-s + 5.47·27-s + 3.00·28-s + 5.85·29-s − 3.61·30-s + ⋯ |
| L(s) = 1 | − 1.58·2-s − 0.934·3-s + 1.50·4-s − 0.447·5-s + 1.47·6-s + 0.377·7-s − 0.790·8-s − 0.127·9-s + 0.707·10-s − 1.40·12-s − 0.660·13-s − 0.597·14-s + 0.417·15-s − 0.249·16-s − 0.820·17-s + 0.201·18-s − 0.283·19-s − 0.670·20-s − 0.353·21-s + 0.932·23-s + 0.738·24-s + 0.200·25-s + 1.04·26-s + 1.05·27-s + 0.566·28-s + 1.08·29-s − 0.660·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.23T + 2T^{2} \) |
| 3 | \( 1 + 1.61T + 3T^{2} \) |
| 13 | \( 1 + 2.38T + 13T^{2} \) |
| 17 | \( 1 + 3.38T + 17T^{2} \) |
| 19 | \( 1 + 1.23T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 - 5.85T + 29T^{2} \) |
| 31 | \( 1 + 1.23T + 31T^{2} \) |
| 37 | \( 1 + 2.76T + 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 - 0.763T + 43T^{2} \) |
| 47 | \( 1 + 7.85T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 4.76T + 59T^{2} \) |
| 61 | \( 1 - 9.23T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 - 0.673T + 71T^{2} \) |
| 73 | \( 1 + 8.61T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 0.326T + 97T^{2} \) |
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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
−8.092582659500586959166230802670, −7.46465613488651781385076081105, −6.71859290254297040118163411549, −6.17792246521172978695981051175, −4.97590174638285796663307971688, −4.55749428408365161686413029873, −3.07106680839664990760492274157, −2.08133672554044300140068799309, −0.920308906870214704398971553972, 0,
0.920308906870214704398971553972, 2.08133672554044300140068799309, 3.07106680839664990760492274157, 4.55749428408365161686413029873, 4.97590174638285796663307971688, 6.17792246521172978695981051175, 6.71859290254297040118163411549, 7.46465613488651781385076081105, 8.092582659500586959166230802670
