| L(s) = 1 | + 1.61·2-s + 0.618·4-s − 2.61·5-s − 4.85·7-s − 2.23·8-s − 4.23·10-s − 1.76·11-s + 3.23·13-s − 7.85·14-s − 4.85·16-s − 17-s + 3·19-s − 1.61·20-s − 2.85·22-s + 4.09·23-s + 1.85·25-s + 5.23·26-s − 3.00·28-s − 8.23·29-s + 4.23·31-s − 3.38·32-s − 1.61·34-s + 12.7·35-s + 0.472·37-s + 4.85·38-s + 5.85·40-s − 6.38·41-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.309·4-s − 1.17·5-s − 1.83·7-s − 0.790·8-s − 1.33·10-s − 0.531·11-s + 0.897·13-s − 2.09·14-s − 1.21·16-s − 0.242·17-s + 0.688·19-s − 0.361·20-s − 0.608·22-s + 0.852·23-s + 0.370·25-s + 1.02·26-s − 0.566·28-s − 1.52·29-s + 0.760·31-s − 0.597·32-s − 0.277·34-s + 2.14·35-s + 0.0776·37-s + 0.787·38-s + 0.925·40-s − 0.996·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 5 | \( 1 + 2.61T + 5T^{2} \) |
| 7 | \( 1 + 4.85T + 7T^{2} \) |
| 11 | \( 1 + 1.76T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 - 4.09T + 23T^{2} \) |
| 29 | \( 1 + 8.23T + 29T^{2} \) |
| 31 | \( 1 - 4.23T + 31T^{2} \) |
| 37 | \( 1 - 0.472T + 37T^{2} \) |
| 41 | \( 1 + 6.38T + 41T^{2} \) |
| 43 | \( 1 - 4.85T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 4.09T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 + 3.14T + 67T^{2} \) |
| 71 | \( 1 - 1.76T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 1.85T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 + 6.32T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−10.90348513533437078225182270530, −9.656793169128973856572005684850, −8.894011005100265104473870482758, −7.66985811191610761036146548009, −6.62670946638433761725637379426, −5.84613034863095469794746401520, −4.62583440749178726044275294502, −3.49921530746824352302871150473, −3.13945368839014386283856455216, 0,
3.13945368839014386283856455216, 3.49921530746824352302871150473, 4.62583440749178726044275294502, 5.84613034863095469794746401520, 6.62670946638433761725637379426, 7.66985811191610761036146548009, 8.894011005100265104473870482758, 9.656793169128973856572005684850, 10.90348513533437078225182270530
