| L(s) = 1 | + 3.22·3-s − 2.53·5-s + 3.87·7-s + 7.41·9-s + 2.34·11-s − 0.837·13-s − 8.17·15-s − 0.958·17-s + 12.5·21-s − 4.36·23-s + 1.41·25-s + 14.2·27-s + 3.46·29-s − 4.63·31-s + 7.57·33-s − 9.82·35-s + 6.22·37-s − 2.70·39-s + 9.39·41-s + 4.47·43-s − 18.7·45-s + 7.17·47-s + 8.04·49-s − 3.09·51-s + 4.23·53-s − 5.94·55-s + 8.43·59-s + ⋯ |
| L(s) = 1 | + 1.86·3-s − 1.13·5-s + 1.46·7-s + 2.47·9-s + 0.707·11-s − 0.232·13-s − 2.10·15-s − 0.232·17-s + 2.73·21-s − 0.911·23-s + 0.282·25-s + 2.73·27-s + 0.643·29-s − 0.833·31-s + 1.31·33-s − 1.66·35-s + 1.02·37-s − 0.432·39-s + 1.46·41-s + 0.682·43-s − 2.79·45-s + 1.04·47-s + 1.14·49-s − 0.432·51-s + 0.581·53-s − 0.801·55-s + 1.09·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.343346159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.343346159\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 3.22T + 3T^{2} \) |
| 5 | \( 1 + 2.53T + 5T^{2} \) |
| 7 | \( 1 - 3.87T + 7T^{2} \) |
| 11 | \( 1 - 2.34T + 11T^{2} \) |
| 13 | \( 1 + 0.837T + 13T^{2} \) |
| 17 | \( 1 + 0.958T + 17T^{2} \) |
| 23 | \( 1 + 4.36T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 + 4.63T + 31T^{2} \) |
| 37 | \( 1 - 6.22T + 37T^{2} \) |
| 41 | \( 1 - 9.39T + 41T^{2} \) |
| 43 | \( 1 - 4.47T + 43T^{2} \) |
| 47 | \( 1 - 7.17T + 47T^{2} \) |
| 53 | \( 1 - 4.23T + 53T^{2} \) |
| 59 | \( 1 - 8.43T + 59T^{2} \) |
| 61 | \( 1 + 6.38T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 8.53T + 71T^{2} \) |
| 73 | \( 1 - 8.90T + 73T^{2} \) |
| 79 | \( 1 + 3.73T + 79T^{2} \) |
| 83 | \( 1 - 2.46T + 83T^{2} \) |
| 89 | \( 1 - 2.28T + 89T^{2} \) |
| 97 | \( 1 - 0.177T + 97T^{2} \) |
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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
−8.043275481358362076164795147000, −7.58492716832477197800341989391, −7.28677740680699535420574776021, −6.06966888317235942368238424238, −4.79126583805711830923184752815, −4.11611005018708047694731663321, −3.88255039045996059715444665933, −2.74461097809267782697635681250, −2.03788718342071426884075781451, −1.09365588572002518888184082354,
1.09365588572002518888184082354, 2.03788718342071426884075781451, 2.74461097809267782697635681250, 3.88255039045996059715444665933, 4.11611005018708047694731663321, 4.79126583805711830923184752815, 6.06966888317235942368238424238, 7.28677740680699535420574776021, 7.58492716832477197800341989391, 8.043275481358362076164795147000
