| L(s) = 1 | + 0.507·2-s + 2.92·3-s − 1.74·4-s − 3.16·5-s + 1.48·6-s − 1.78·7-s − 1.89·8-s + 5.57·9-s − 1.60·10-s − 1.02·11-s − 5.10·12-s + 13-s − 0.905·14-s − 9.25·15-s + 2.52·16-s − 1.61·17-s + 2.82·18-s − 7.50·19-s + 5.50·20-s − 5.22·21-s − 0.519·22-s − 4.04·23-s − 5.56·24-s + 4.99·25-s + 0.507·26-s + 7.54·27-s + 3.11·28-s + ⋯ |
| L(s) = 1 | + 0.358·2-s + 1.69·3-s − 0.871·4-s − 1.41·5-s + 0.606·6-s − 0.674·7-s − 0.671·8-s + 1.85·9-s − 0.507·10-s − 0.308·11-s − 1.47·12-s + 0.277·13-s − 0.242·14-s − 2.39·15-s + 0.630·16-s − 0.391·17-s + 0.666·18-s − 1.72·19-s + 1.23·20-s − 1.14·21-s − 0.110·22-s − 0.844·23-s − 1.13·24-s + 0.999·25-s + 0.0995·26-s + 1.45·27-s + 0.587·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 0.507T + 2T^{2} \) |
| 3 | \( 1 - 2.92T + 3T^{2} \) |
| 5 | \( 1 + 3.16T + 5T^{2} \) |
| 7 | \( 1 + 1.78T + 7T^{2} \) |
| 11 | \( 1 + 1.02T + 11T^{2} \) |
| 17 | \( 1 + 1.61T + 17T^{2} \) |
| 19 | \( 1 + 7.50T + 19T^{2} \) |
| 23 | \( 1 + 4.04T + 23T^{2} \) |
| 29 | \( 1 + 1.84T + 29T^{2} \) |
| 31 | \( 1 + 9.98T + 31T^{2} \) |
| 37 | \( 1 + 1.79T + 37T^{2} \) |
| 41 | \( 1 + 6.85T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 - 9.45T + 47T^{2} \) |
| 53 | \( 1 + 5.61T + 53T^{2} \) |
| 59 | \( 1 + 5.43T + 59T^{2} \) |
| 61 | \( 1 - 2.69T + 61T^{2} \) |
| 67 | \( 1 - 4.00T + 67T^{2} \) |
| 71 | \( 1 + 9.89T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 83 | \( 1 - 17.5T + 83T^{2} \) |
| 89 | \( 1 - 1.77T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−9.102966454010256779108202743629, −8.787506698438745547781708298297, −7.980036899256230947569307532339, −7.39831792157530304713622956474, −6.17881071785864956969189171432, −4.66888703097718794632140453467, −3.77409185744132578592384005645, −3.58980807821395023409437200834, −2.28184809886301369365056067918, 0,
2.28184809886301369365056067918, 3.58980807821395023409437200834, 3.77409185744132578592384005645, 4.66888703097718794632140453467, 6.17881071785864956969189171432, 7.39831792157530304713622956474, 7.980036899256230947569307532339, 8.787506698438745547781708298297, 9.102966454010256779108202743629
