| L(s) = 1 | − 9.95e5·5-s − 1.18e9·7-s + 3.52e10·11-s − 8.44e11·13-s − 1.17e13·17-s − 1.89e13·19-s + 2.72e14·23-s − 4.75e14·25-s − 1.38e15·29-s − 1.93e15·31-s + 1.17e15·35-s − 5.29e16·37-s − 1.37e17·41-s − 1.40e16·43-s + 1.53e17·47-s + 8.34e17·49-s + 1.04e18·53-s − 3.50e16·55-s − 4.07e18·59-s + 3.59e18·61-s + 8.41e17·65-s − 6.22e16·67-s − 3.80e19·71-s + 5.06e19·73-s − 4.15e19·77-s − 2.16e19·79-s − 1.97e20·83-s + ⋯ |
| L(s) = 1 | − 0.0456·5-s − 1.57·7-s + 0.409·11-s − 1.69·13-s − 1.41·17-s − 0.708·19-s + 1.37·23-s − 0.997·25-s − 0.610·29-s − 0.424·31-s + 0.0720·35-s − 1.81·37-s − 1.59·41-s − 0.0988·43-s + 0.424·47-s + 1.49·49-s + 0.817·53-s − 0.0186·55-s − 1.03·59-s + 0.644·61-s + 0.0774·65-s − 0.00417·67-s − 1.38·71-s + 1.37·73-s − 0.646·77-s − 0.256·79-s − 1.39·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(0.1534442125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1534442125\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 + 9.95e5T + 4.76e14T^{2} \) |
| 7 | \( 1 + 1.18e9T + 5.58e17T^{2} \) |
| 11 | \( 1 - 3.52e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 8.44e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 1.17e13T + 6.90e25T^{2} \) |
| 19 | \( 1 + 1.89e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 2.72e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 1.38e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 1.93e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 5.29e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 1.37e17T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.40e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 1.53e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.04e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 4.07e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 3.59e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 6.22e16T + 2.22e38T^{2} \) |
| 71 | \( 1 + 3.80e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 5.06e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 2.16e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.97e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 9.11e19T + 8.65e40T^{2} \) |
| 97 | \( 1 - 9.35e20T + 5.27e41T^{2} \) |
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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
−10.55524349291798596880703965275, −9.589074262604815728882846510926, −8.829198314314158100469315027565, −7.14869423645462285138627761957, −6.61905811600708808342239316212, −5.25748305429894699522943793211, −4.02579070002669671107599207679, −2.94178127192979597863963596243, −1.93903778956629398116596090813, −0.15213978301231975649587283637,
0.15213978301231975649587283637, 1.93903778956629398116596090813, 2.94178127192979597863963596243, 4.02579070002669671107599207679, 5.25748305429894699522943793211, 6.61905811600708808342239316212, 7.14869423645462285138627761957, 8.829198314314158100469315027565, 9.589074262604815728882846510926, 10.55524349291798596880703965275
