| L(s) = 1 | − 3.06·2-s + 3·3-s + 1.36·4-s − 15.4·5-s − 9.18·6-s + 12.8·7-s + 20.3·8-s + 9·9-s + 47.2·10-s + 11·11-s + 4.09·12-s + 61.0·13-s − 39.2·14-s − 46.2·15-s − 73.0·16-s − 94.6·17-s − 27.5·18-s − 98.3·19-s − 21.0·20-s + 38.4·21-s − 33.6·22-s + 18.7·23-s + 60.9·24-s + 112.·25-s − 186.·26-s + 27·27-s + 17.5·28-s + ⋯ |
| L(s) = 1 | − 1.08·2-s + 0.577·3-s + 0.170·4-s − 1.37·5-s − 0.624·6-s + 0.692·7-s + 0.897·8-s + 0.333·9-s + 1.49·10-s + 0.301·11-s + 0.0985·12-s + 1.30·13-s − 0.749·14-s − 0.796·15-s − 1.14·16-s − 1.35·17-s − 0.360·18-s − 1.18·19-s − 0.235·20-s + 0.399·21-s − 0.326·22-s + 0.169·23-s + 0.518·24-s + 0.903·25-s − 1.40·26-s + 0.192·27-s + 0.118·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 3.06T + 8T^{2} \) |
| 5 | \( 1 + 15.4T + 125T^{2} \) |
| 7 | \( 1 - 12.8T + 343T^{2} \) |
| 13 | \( 1 - 61.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 94.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 98.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 18.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 259.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 118.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 243.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 444.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 198.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 22.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 624.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 681.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 382.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 861.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 589.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 721.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.61e3T + 9.12e5T^{2} \) |
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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
−8.384519670855691963351727727063, −8.199689309671441208298828363548, −7.04752132118004600583264445339, −6.51608844600294498338819609699, −4.76937914262149434731627852680, −4.29244907657869708106013481848, −3.46015455347811779801334087148, −2.06487897655629967338406842873, −1.07142867768308494539270573611, 0,
1.07142867768308494539270573611, 2.06487897655629967338406842873, 3.46015455347811779801334087148, 4.29244907657869708106013481848, 4.76937914262149434731627852680, 6.51608844600294498338819609699, 7.04752132118004600583264445339, 8.199689309671441208298828363548, 8.384519670855691963351727727063
