L(s) = 1 | + 2.16·5-s − 2.64i·7-s − 4.55·13-s + 0.979i·19-s − 6i·23-s − 0.291·25-s − 5.74i·35-s − 7.00·49-s − 14.4i·59-s − 15.6·61-s − 9.87·65-s − 15.8i·71-s − 5.29i·79-s + 1.40i·83-s + 12.0i·91-s + ⋯ |
L(s) = 1 | + 0.970·5-s − 0.999i·7-s − 1.26·13-s + 0.224i·19-s − 1.25i·23-s − 0.0583·25-s − 0.970i·35-s − 49-s − 1.87i·59-s − 1.99·61-s − 1.22·65-s − 1.88i·71-s − 0.595i·79-s + 0.153i·83-s + 1.26i·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.215027629\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.215027629\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
good | 5 | \( 1 - 2.16T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4.55T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 0.979iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 14.4iT - 59T^{2} \) |
| 61 | \( 1 + 15.6T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 15.8iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 5.29iT - 79T^{2} \) |
| 83 | \( 1 - 1.40iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.010088833610920260965459342276, −7.47516251784418575054985429983, −6.61683594160161620479691424151, −6.10067829703490184674252945341, −5.03384397825934031104576656439, −4.54975056668956547417748182751, −3.48795091308810697470708076986, −2.48734906661981848553451924962, −1.64712820473225209484076140818, −0.31511251794058783961764581973,
1.50641882092080205563680983455, 2.36353364053478344928124147467, 3.01558116658420293503477746332, 4.27467357553202860653187387594, 5.24930798627245230028300378900, 5.62273890272500780600559326941, 6.40377594234714886705949677099, 7.27275189219451302055615714899, 7.947657376699942392036489997587, 8.951651766852160218405557864697