| L(s) = 1 | + 2.83·2-s + 3·3-s + 0.0281·4-s + 13.6·5-s + 8.50·6-s − 2.14·7-s − 22.5·8-s + 9·9-s + 38.6·10-s − 11·11-s + 0.0845·12-s − 23.9·13-s − 6.06·14-s + 40.9·15-s − 64.2·16-s − 93.1·17-s + 25.5·18-s + 24.5·19-s + 0.384·20-s − 6.42·21-s − 31.1·22-s + 161.·23-s − 67.7·24-s + 61.5·25-s − 67.9·26-s + 27·27-s − 0.0603·28-s + ⋯ |
| L(s) = 1 | + 1.00·2-s + 0.577·3-s + 0.00352·4-s + 1.22·5-s + 0.578·6-s − 0.115·7-s − 0.998·8-s + 0.333·9-s + 1.22·10-s − 0.301·11-s + 0.00203·12-s − 0.511·13-s − 0.115·14-s + 0.705·15-s − 1.00·16-s − 1.32·17-s + 0.333·18-s + 0.296·19-s + 0.00430·20-s − 0.0667·21-s − 0.302·22-s + 1.46·23-s − 0.576·24-s + 0.492·25-s − 0.512·26-s + 0.192·27-s − 0.000407·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 - 2.83T + 8T^{2} \) |
| 5 | \( 1 - 13.6T + 125T^{2} \) |
| 7 | \( 1 + 2.14T + 343T^{2} \) |
| 13 | \( 1 + 23.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 93.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 24.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 246.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 44.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 425.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 91.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 542.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 66.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 85.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 530.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 373.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 409.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 628.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 624.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.44e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.06e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 79.5T + 9.12e5T^{2} \) |
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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
−8.713517845578276067092386555254, −7.40760384155697395032816581025, −6.69409859526503047433465032619, −5.78143318523122491227477216879, −5.16425704688879082409307448240, −4.39912022727799276284403876768, −3.34445967837915039310028368420, −2.57134955791613744922169577853, −1.70798032991902734045144598442, 0,
1.70798032991902734045144598442, 2.57134955791613744922169577853, 3.34445967837915039310028368420, 4.39912022727799276284403876768, 5.16425704688879082409307448240, 5.78143318523122491227477216879, 6.69409859526503047433465032619, 7.40760384155697395032816581025, 8.713517845578276067092386555254
