| L(s) = 1 | − 0.618·2-s − 1.61·4-s − 0.381·5-s + 1.85·7-s + 2.23·8-s + 0.236·10-s − 6.23·11-s − 1.23·13-s − 1.14·14-s + 1.85·16-s − 17-s + 3·19-s + 0.618·20-s + 3.85·22-s − 7.09·23-s − 4.85·25-s + 0.763·26-s − 3·28-s − 3.76·29-s − 0.236·31-s − 5.61·32-s + 0.618·34-s − 0.708·35-s − 8.47·37-s − 1.85·38-s − 0.854·40-s − 8.61·41-s + ⋯ |
| L(s) = 1 | − 0.437·2-s − 0.809·4-s − 0.170·5-s + 0.700·7-s + 0.790·8-s + 0.0746·10-s − 1.88·11-s − 0.342·13-s − 0.306·14-s + 0.463·16-s − 0.242·17-s + 0.688·19-s + 0.138·20-s + 0.821·22-s − 1.47·23-s − 0.970·25-s + 0.149·26-s − 0.566·28-s − 0.698·29-s − 0.0423·31-s − 0.993·32-s + 0.105·34-s − 0.119·35-s − 1.39·37-s − 0.300·38-s − 0.135·40-s − 1.34·41-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 | \( 1 + 0.618T + 2T^{2} \) |
| 5 | \( 1 + 0.381T + 5T^{2} \) |
| 7 | \( 1 - 1.85T + 7T^{2} \) |
| 11 | \( 1 + 6.23T + 11T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 7.09T + 23T^{2} \) |
| 29 | \( 1 + 3.76T + 29T^{2} \) |
| 31 | \( 1 + 0.236T + 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 + 8.61T + 41T^{2} \) |
| 43 | \( 1 + 1.85T + 43T^{2} \) |
| 47 | \( 1 - 4.32T + 47T^{2} \) |
| 53 | \( 1 - 5.32T + 53T^{2} \) |
| 59 | \( 1 - 7.09T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 + 9.85T + 67T^{2} \) |
| 71 | \( 1 - 6.23T + 71T^{2} \) |
| 73 | \( 1 + 2.52T + 73T^{2} \) |
| 79 | \( 1 + 4.85T + 79T^{2} \) |
| 83 | \( 1 - 6.52T + 83T^{2} \) |
| 89 | \( 1 - 9.32T + 89T^{2} \) |
| 97 | \( 1 - 0.909T + 97T^{2} \) |
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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
−10.30861467391288214749127082394, −9.916728527331014851382104552332, −8.630960899456274033118945197119, −7.988638855078182716717421417526, −7.34687233561946531043195396841, −5.54846567537103418410337763388, −4.95684774125784851642822673312, −3.73077871325009449494545283471, −2.05437589228643690193884427190, 0,
2.05437589228643690193884427190, 3.73077871325009449494545283471, 4.95684774125784851642822673312, 5.54846567537103418410337763388, 7.34687233561946531043195396841, 7.988638855078182716717421417526, 8.630960899456274033118945197119, 9.916728527331014851382104552332, 10.30861467391288214749127082394
