| L(s) = 1 | − 2.50·2-s + 1.55·3-s + 4.29·4-s − 3.66·5-s − 3.90·6-s + 4.73·7-s − 5.76·8-s − 0.575·9-s + 9.19·10-s + 5.22·11-s + 6.69·12-s − 3.90·13-s − 11.8·14-s − 5.70·15-s + 5.87·16-s − 3.68·17-s + 1.44·18-s − 5.02·19-s − 15.7·20-s + 7.37·21-s − 13.1·22-s − 0.531·23-s − 8.97·24-s + 8.41·25-s + 9.78·26-s − 5.56·27-s + 20.3·28-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 0.899·3-s + 2.14·4-s − 1.63·5-s − 1.59·6-s + 1.79·7-s − 2.03·8-s − 0.191·9-s + 2.90·10-s + 1.57·11-s + 1.93·12-s − 1.08·13-s − 3.17·14-s − 1.47·15-s + 1.46·16-s − 0.893·17-s + 0.340·18-s − 1.15·19-s − 3.51·20-s + 1.61·21-s − 2.79·22-s − 0.110·23-s − 1.83·24-s + 1.68·25-s + 1.91·26-s − 1.07·27-s + 3.84·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 37 | \( 1 + T \) |
| 109 | \( 1 + T \) |
| good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 3 | \( 1 - 1.55T + 3T^{2} \) |
| 5 | \( 1 + 3.66T + 5T^{2} \) |
| 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 - 5.22T + 11T^{2} \) |
| 13 | \( 1 + 3.90T + 13T^{2} \) |
| 17 | \( 1 + 3.68T + 17T^{2} \) |
| 19 | \( 1 + 5.02T + 19T^{2} \) |
| 23 | \( 1 + 0.531T + 23T^{2} \) |
| 29 | \( 1 - 7.93T + 29T^{2} \) |
| 31 | \( 1 - 9.09T + 31T^{2} \) |
| 41 | \( 1 + 9.30T + 41T^{2} \) |
| 43 | \( 1 + 1.44T + 43T^{2} \) |
| 47 | \( 1 + 4.67T + 47T^{2} \) |
| 53 | \( 1 + 9.23T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 - 7.43T + 61T^{2} \) |
| 67 | \( 1 - 6.71T + 67T^{2} \) |
| 71 | \( 1 + 3.82T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 - 3.11T + 79T^{2} \) |
| 83 | \( 1 + 6.20T + 83T^{2} \) |
| 89 | \( 1 + 2.99T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−8.198098559325052850330860535956, −7.88388824451330446602508982519, −6.99766525430574856753953884969, −6.49214129323379457974875302232, −4.67451197408775782796593021715, −4.31194248628931202416587303320, −3.07668890495502892368872042957, −2.14820628053939124891354447983, −1.30171591971930536460937028209, 0,
1.30171591971930536460937028209, 2.14820628053939124891354447983, 3.07668890495502892368872042957, 4.31194248628931202416587303320, 4.67451197408775782796593021715, 6.49214129323379457974875302232, 6.99766525430574856753953884969, 7.88388824451330446602508982519, 8.198098559325052850330860535956
