| L(s) = 1 | + 4.12·5-s + 3.56·7-s + 3.56·11-s + 5.56·13-s − 0.561·17-s − 4·19-s − 5·23-s + 12·25-s + 5.12·29-s − 4·31-s + 14.6·35-s − 3.68·37-s − 9.56·41-s + 2.56·43-s + 7.12·47-s + 5.68·49-s − 9.12·53-s + 14.6·55-s − 6.43·59-s + 61-s + 22.9·65-s + 3.31·67-s + 13.9·71-s + 16.3·73-s + 12.6·77-s + 12.9·79-s − 10.8·83-s + ⋯ |
| L(s) = 1 | + 1.84·5-s + 1.34·7-s + 1.07·11-s + 1.54·13-s − 0.136·17-s − 0.917·19-s − 1.04·23-s + 2.40·25-s + 0.951·29-s − 0.718·31-s + 2.48·35-s − 0.605·37-s − 1.49·41-s + 0.390·43-s + 1.03·47-s + 0.812·49-s − 1.25·53-s + 1.98·55-s − 0.838·59-s + 0.128·61-s + 2.84·65-s + 0.405·67-s + 1.65·71-s + 1.91·73-s + 1.44·77-s + 1.45·79-s − 1.18·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.400379802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.400379802\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 - 4.12T + 5T^{2} \) |
| 7 | \( 1 - 3.56T + 7T^{2} \) |
| 11 | \( 1 - 3.56T + 11T^{2} \) |
| 13 | \( 1 - 5.56T + 13T^{2} \) |
| 17 | \( 1 + 0.561T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 5T + 23T^{2} \) |
| 29 | \( 1 - 5.12T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 3.68T + 37T^{2} \) |
| 41 | \( 1 + 9.56T + 41T^{2} \) |
| 43 | \( 1 - 2.56T + 43T^{2} \) |
| 47 | \( 1 - 7.12T + 47T^{2} \) |
| 53 | \( 1 + 9.12T + 53T^{2} \) |
| 59 | \( 1 + 6.43T + 59T^{2} \) |
| 67 | \( 1 - 3.31T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 + 3.43T + 97T^{2} \) |
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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
−7.964646523231635693916031174328, −6.65462672221849624973221040893, −6.47850698368629031830395626661, −5.71781705454767233506905182249, −5.12183477599327489484243505990, −4.30998827016662809419079898599, −3.53680856380664299390535945535, −2.27847953041382123107512835709, −1.70758647325552011171081640599, −1.17759715663722635125754418662,
1.17759715663722635125754418662, 1.70758647325552011171081640599, 2.27847953041382123107512835709, 3.53680856380664299390535945535, 4.30998827016662809419079898599, 5.12183477599327489484243505990, 5.71781705454767233506905182249, 6.47850698368629031830395626661, 6.65462672221849624973221040893, 7.964646523231635693916031174328
