| L(s) = 1 | + 5-s − 3.64·7-s − 3.64·11-s − 1.64·13-s − 0.645·17-s − 2.64·19-s − 6.29·23-s + 25-s + 0.354·29-s + 2.64·31-s − 3.64·35-s + 5.29·37-s + 5.64·41-s + 8.93·43-s + 5.29·47-s + 6.29·49-s + 11.9·53-s − 3.64·55-s + 1.64·59-s − 13.5·61-s − 1.64·65-s + 6·67-s + 7.64·71-s + 14.9·73-s + 13.2·77-s − 12.6·79-s − 5·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.37·7-s − 1.09·11-s − 0.456·13-s − 0.156·17-s − 0.606·19-s − 1.31·23-s + 0.200·25-s + 0.0657·29-s + 0.475·31-s − 0.616·35-s + 0.869·37-s + 0.881·41-s + 1.36·43-s + 0.771·47-s + 0.898·49-s + 1.63·53-s − 0.491·55-s + 0.214·59-s − 1.73·61-s − 0.204·65-s + 0.733·67-s + 0.907·71-s + 1.74·73-s + 1.51·77-s − 1.42·79-s − 0.548·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.144362068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.144362068\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 3.64T + 7T^{2} \) |
| 11 | \( 1 + 3.64T + 11T^{2} \) |
| 13 | \( 1 + 1.64T + 13T^{2} \) |
| 17 | \( 1 + 0.645T + 17T^{2} \) |
| 19 | \( 1 + 2.64T + 19T^{2} \) |
| 23 | \( 1 + 6.29T + 23T^{2} \) |
| 29 | \( 1 - 0.354T + 29T^{2} \) |
| 31 | \( 1 - 2.64T + 31T^{2} \) |
| 37 | \( 1 - 5.29T + 37T^{2} \) |
| 41 | \( 1 - 5.64T + 41T^{2} \) |
| 43 | \( 1 - 8.93T + 43T^{2} \) |
| 47 | \( 1 - 5.29T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 - 1.64T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 - 6T + 67T^{2} \) |
| 71 | \( 1 - 7.64T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 5T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−8.342509921805721901857345501757, −7.63401025837079178462894069046, −6.87555070510443013174835379715, −6.06214530184113424180862402954, −5.68227133861035876134870031138, −4.58937262026151251635041753194, −3.80002371130049859718052740118, −2.70905632307146620253206277415, −2.27403980351110964475763902809, −0.56666097790651523729993303139,
0.56666097790651523729993303139, 2.27403980351110964475763902809, 2.70905632307146620253206277415, 3.80002371130049859718052740118, 4.58937262026151251635041753194, 5.68227133861035876134870031138, 6.06214530184113424180862402954, 6.87555070510443013174835379715, 7.63401025837079178462894069046, 8.342509921805721901857345501757
