| L(s) = 1 | + 1.61·2-s − 2.23·3-s + 0.618·4-s − 3.61·6-s + 1.23·7-s − 2.23·8-s + 2.00·9-s − 0.763·11-s − 1.38·12-s − 3·13-s + 2.00·14-s − 4.85·16-s − 5.23·17-s + 3.23·18-s − 2·19-s − 2.76·21-s − 1.23·22-s − 23-s + 5.00·24-s − 4.85·26-s + 2.23·27-s + 0.763·28-s − 3·29-s − 6.70·31-s − 3.38·32-s + 1.70·33-s − 8.47·34-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 1.29·3-s + 0.309·4-s − 1.47·6-s + 0.467·7-s − 0.790·8-s + 0.666·9-s − 0.230·11-s − 0.398·12-s − 0.832·13-s + 0.534·14-s − 1.21·16-s − 1.26·17-s + 0.762·18-s − 0.458·19-s − 0.603·21-s − 0.263·22-s − 0.208·23-s + 1.02·24-s − 0.951·26-s + 0.430·27-s + 0.144·28-s − 0.557·29-s − 1.20·31-s − 0.597·32-s + 0.297·33-s − 1.45·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 3 | \( 1 + 2.23T + 3T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 + 3.23T + 37T^{2} \) |
| 41 | \( 1 - 5.47T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 2.23T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 + 2.47T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 7.23T + 67T^{2} \) |
| 71 | \( 1 - 7.76T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 - 6.94T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 + 1.52T + 89T^{2} \) |
| 97 | \( 1 + 4.29T + 97T^{2} \) |
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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
−10.76009312718436085231162643313, −9.547864028753442484523812998478, −8.526534161097957062624098187443, −7.16107845661655673232901680953, −6.32034116706534023990351271627, −5.41226095038832743760400764873, −4.85469906029089226711569754979, −3.93241212986199430247095182147, −2.32708019964692189665792439050, 0,
2.32708019964692189665792439050, 3.93241212986199430247095182147, 4.85469906029089226711569754979, 5.41226095038832743760400764873, 6.32034116706534023990351271627, 7.16107845661655673232901680953, 8.526534161097957062624098187443, 9.547864028753442484523812998478, 10.76009312718436085231162643313
