| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 11-s + 13-s + 15-s − 17-s + 19-s − 21-s − 23-s + 25-s + 27-s + 31-s + 33-s − 35-s − 37-s + 39-s − 41-s + 43-s + 45-s + 47-s + 49-s − 51-s + 53-s + 55-s + 57-s + ⋯ |
| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 11-s + 13-s + 15-s − 17-s + 19-s − 21-s − 23-s + 25-s + 27-s + 31-s + 33-s − 35-s − 37-s + 39-s − 41-s + 43-s + 45-s + 47-s + 49-s − 51-s + 53-s + 55-s + 57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.884518775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.884518775\) |
| \(L(1)\) |
\(\approx\) |
\(1.750137330\) |
| \(L(1)\) |
\(\approx\) |
\(1.750137330\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−29.10728884062669449984644030521, −28.05817943106239695976732655940, −26.59677823869733120585123080170, −25.91057950875139359531491722626, −25.094720800369474213100059780532, −24.29455239315401670929083628302, −22.58255821231777735983801196344, −21.839864950721796025022224864965, −20.63057804260674935785541900476, −19.83502358996123754453981580244, −18.747444327813632650386305438473, −17.7010932101502087834912320818, −16.31970268674881718170052687995, −15.35734320320145801398011847233, −13.82937879351773359600082980971, −13.59278038210108867253298124701, −12.211062797860126232968613635788, −10.411402146226654700457772204130, −9.415397688391975929238262604479, −8.701173952921560494904822111532, −6.96956393553234557071638797738, −6.017382235992170490422307344071, −4.08103659546516473538382287793, −2.868371820651405764455340610178, −1.452918860738990389442431257107,
1.452918860738990389442431257107, 2.868371820651405764455340610178, 4.08103659546516473538382287793, 6.017382235992170490422307344071, 6.96956393553234557071638797738, 8.701173952921560494904822111532, 9.415397688391975929238262604479, 10.411402146226654700457772204130, 12.211062797860126232968613635788, 13.59278038210108867253298124701, 13.82937879351773359600082980971, 15.35734320320145801398011847233, 16.31970268674881718170052687995, 17.7010932101502087834912320818, 18.747444327813632650386305438473, 19.83502358996123754453981580244, 20.63057804260674935785541900476, 21.839864950721796025022224864965, 22.58255821231777735983801196344, 24.29455239315401670929083628302, 25.094720800369474213100059780532, 25.91057950875139359531491722626, 26.59677823869733120585123080170, 28.05817943106239695976732655940, 29.10728884062669449984644030521
