| L(s) = 1 | − 2.11·2-s + 3-s + 2.47·4-s − 2.11·6-s − 5.11·7-s − 1.00·8-s + 9-s − 4.58·11-s + 2.47·12-s + 2.58·13-s + 10.8·14-s − 2.83·16-s − 2.18·17-s − 2.11·18-s + 0.527·19-s − 5.11·21-s + 9.70·22-s + 5.70·23-s − 1.00·24-s − 5.47·26-s + 27-s − 12.6·28-s + 2.06·29-s + 5.83·31-s + 7.98·32-s − 4.58·33-s + 4.62·34-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 0.577·3-s + 1.23·4-s − 0.863·6-s − 1.93·7-s − 0.353·8-s + 0.333·9-s − 1.38·11-s + 0.713·12-s + 0.717·13-s + 2.89·14-s − 0.707·16-s − 0.530·17-s − 0.498·18-s + 0.120·19-s − 1.11·21-s + 2.06·22-s + 1.18·23-s − 0.204·24-s − 1.07·26-s + 0.192·27-s − 2.39·28-s + 0.382·29-s + 1.04·31-s + 1.41·32-s − 0.798·33-s + 0.793·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 + 2.11T + 2T^{2} \) |
| 7 | \( 1 + 5.11T + 7T^{2} \) |
| 11 | \( 1 + 4.58T + 11T^{2} \) |
| 13 | \( 1 - 2.58T + 13T^{2} \) |
| 17 | \( 1 + 2.18T + 17T^{2} \) |
| 19 | \( 1 - 0.527T + 19T^{2} \) |
| 23 | \( 1 - 5.70T + 23T^{2} \) |
| 29 | \( 1 - 2.06T + 29T^{2} \) |
| 31 | \( 1 - 5.83T + 31T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 41 | \( 1 - 2.39T + 41T^{2} \) |
| 43 | \( 1 + 7.15T + 43T^{2} \) |
| 47 | \( 1 + 8.77T + 47T^{2} \) |
| 53 | \( 1 - 8.10T + 53T^{2} \) |
| 61 | \( 1 - 9.00T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 - 4.12T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 3.87T + 79T^{2} \) |
| 83 | \( 1 - 0.737T + 83T^{2} \) |
| 89 | \( 1 + 8.54T + 89T^{2} \) |
| 97 | \( 1 - 4.10T + 97T^{2} \) |
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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
−8.262257040385384260673979186954, −7.42302861821341378264613737555, −6.75312941092329636085435303853, −6.24662792703790980199577693046, −5.08884907341439110173867796455, −3.95095398473482245565017734228, −2.89570991874184602495455526681, −2.54512362031839136224242614360, −1.05562001539645927867492300210, 0,
1.05562001539645927867492300210, 2.54512362031839136224242614360, 2.89570991874184602495455526681, 3.95095398473482245565017734228, 5.08884907341439110173867796455, 6.24662792703790980199577693046, 6.75312941092329636085435303853, 7.42302861821341378264613737555, 8.262257040385384260673979186954
