| L(s) = 1 | + 252.·3-s + 2.01e3·5-s + 3.03e3·7-s + 4.41e4·9-s − 4.24e4·11-s − 7.41e4·13-s + 5.10e5·15-s − 6.07e5·17-s − 1.64e5·19-s + 7.68e5·21-s + 2.08e6·23-s + 2.12e6·25-s + 6.19e6·27-s + 1.87e6·29-s − 6.69e5·31-s − 1.07e7·33-s + 6.13e6·35-s − 5.06e6·37-s − 1.87e7·39-s − 1.46e7·41-s − 1.15e7·43-s + 8.92e7·45-s + 3.35e7·47-s − 3.11e7·49-s − 1.53e8·51-s − 2.03e7·53-s − 8.58e7·55-s + ⋯ |
| L(s) = 1 | + 1.80·3-s + 1.44·5-s + 0.478·7-s + 2.24·9-s − 0.875·11-s − 0.720·13-s + 2.60·15-s − 1.76·17-s − 0.290·19-s + 0.861·21-s + 1.55·23-s + 1.08·25-s + 2.24·27-s + 0.492·29-s − 0.130·31-s − 1.57·33-s + 0.691·35-s − 0.443·37-s − 1.29·39-s − 0.808·41-s − 0.517·43-s + 3.24·45-s + 1.00·47-s − 0.771·49-s − 3.17·51-s − 0.355·53-s − 1.26·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.095318164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.095318164\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 252.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.01e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 3.03e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.24e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.41e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 6.07e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.64e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.08e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.87e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.69e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 5.06e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.46e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.15e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.35e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.03e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.19e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.81e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.01e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.11e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 6.82e6T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.23e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.37e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.21e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.11e9T + 7.60e17T^{2} \) |
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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
−14.61196508332687456241825280383, −13.57011899031104444858753122405, −13.01496005788793699353265360045, −10.53920138643421407714550341994, −9.392791538537532033175987673594, −8.490335207057929231208692765928, −6.97656415521782590375495518325, −4.83661260282335582137713640717, −2.72762085006925955836623305090, −1.87636848071045217714746157721,
1.87636848071045217714746157721, 2.72762085006925955836623305090, 4.83661260282335582137713640717, 6.97656415521782590375495518325, 8.490335207057929231208692765928, 9.392791538537532033175987673594, 10.53920138643421407714550341994, 13.01496005788793699353265360045, 13.57011899031104444858753122405, 14.61196508332687456241825280383
