| L(s) = 1 | + 5-s − 4.89·7-s − 4.89·11-s + 3·13-s + 5·17-s + 4.89·19-s − 4.89·23-s − 4·25-s + 5·29-s − 4.89·35-s − 5·37-s − 2·41-s + 4.89·43-s + 9.79·47-s + 16.9·49-s + 2·53-s − 4.89·55-s + 9.79·59-s − 13·61-s + 3·65-s + 4.89·67-s + 4.89·71-s + 3·73-s + 23.9·77-s + 14.6·79-s + 9.79·83-s + 5·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.85·7-s − 1.47·11-s + 0.832·13-s + 1.21·17-s + 1.12·19-s − 1.02·23-s − 0.800·25-s + 0.928·29-s − 0.828·35-s − 0.821·37-s − 0.312·41-s + 0.747·43-s + 1.42·47-s + 2.42·49-s + 0.274·53-s − 0.660·55-s + 1.27·59-s − 1.66·61-s + 0.372·65-s + 0.598·67-s + 0.581·71-s + 0.351·73-s + 2.73·77-s + 1.65·79-s + 1.07·83-s + 0.542·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.345847668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.345847668\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + 4.89T + 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 19 | \( 1 - 4.89T + 19T^{2} \) |
| 23 | \( 1 + 4.89T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4.89T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 9.79T + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 - 4.89T + 67T^{2} \) |
| 71 | \( 1 - 4.89T + 71T^{2} \) |
| 73 | \( 1 - 3T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 9.79T + 83T^{2} \) |
| 89 | \( 1 - 13T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.038405474590847809098253566237, −8.019664669712894958738910080091, −7.41464839354200710350958445994, −6.42041713342509422536734630160, −5.81634867419905918323451262839, −5.25325808955941660810469617642, −3.78727183164310103357149454236, −3.19914448931080631562603241290, −2.31855529708249890447889985474, −0.70774184143114507895356192899,
0.70774184143114507895356192899, 2.31855529708249890447889985474, 3.19914448931080631562603241290, 3.78727183164310103357149454236, 5.25325808955941660810469617642, 5.81634867419905918323451262839, 6.42041713342509422536734630160, 7.41464839354200710350958445994, 8.019664669712894958738910080091, 9.038405474590847809098253566237
