| L(s) = 1 | − 0.0881·2-s + 3·3-s − 7.99·4-s − 5·5-s − 0.264·6-s − 20.0·7-s + 1.41·8-s + 9·9-s + 0.440·10-s − 23.9·12-s + 19.8·13-s + 1.77·14-s − 15·15-s + 63.8·16-s + 7.50·17-s − 0.793·18-s + 27.5·19-s + 39.9·20-s − 60.2·21-s − 27.3·23-s + 4.23·24-s + 25·25-s − 1.74·26-s + 27·27-s + 160.·28-s − 98.3·29-s + 1.32·30-s + ⋯ |
| L(s) = 1 | − 0.0311·2-s + 0.577·3-s − 0.999·4-s − 0.447·5-s − 0.0179·6-s − 1.08·7-s + 0.0623·8-s + 0.333·9-s + 0.0139·10-s − 0.576·12-s + 0.422·13-s + 0.0338·14-s − 0.258·15-s + 0.997·16-s + 0.107·17-s − 0.0103·18-s + 0.332·19-s + 0.446·20-s − 0.626·21-s − 0.248·23-s + 0.0359·24-s + 0.200·25-s − 0.0131·26-s + 0.192·27-s + 1.08·28-s − 0.629·29-s + 0.00804·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.0881T + 8T^{2} \) |
| 7 | \( 1 + 20.0T + 343T^{2} \) |
| 13 | \( 1 - 19.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 7.50T + 4.91e3T^{2} \) |
| 19 | \( 1 - 27.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 27.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 98.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 88.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 53.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 197.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 37.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 528.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 306.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 69.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 779.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 911.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 100.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 173.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 183.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.14e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.16e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.20e3T + 9.12e5T^{2} \) |
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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
−8.576712562193838920500461407779, −7.891017011625802491851814626045, −7.05665970174207941360582931939, −6.09225921027473624425431942479, −5.19013428714768207409532449051, −4.07150746441515577809067854367, −3.61175897445654334245236361920, −2.64770600753337033924852657964, −1.08156549464926757125270584885, 0,
1.08156549464926757125270584885, 2.64770600753337033924852657964, 3.61175897445654334245236361920, 4.07150746441515577809067854367, 5.19013428714768207409532449051, 6.09225921027473624425431942479, 7.05665970174207941360582931939, 7.891017011625802491851814626045, 8.576712562193838920500461407779
