| L(s) = 1 | − 0.232·3-s − 1.79e3·5-s − 2.40e3·7-s − 1.96e4·9-s − 1.74e4·11-s − 1.22e5·13-s + 416.·15-s + 3.31e5·17-s − 7.61e5·19-s + 557.·21-s − 1.23e6·23-s + 1.25e6·25-s + 9.14e3·27-s + 6.34e5·29-s + 5.38e6·31-s + 4.04e3·33-s + 4.30e6·35-s − 3.03e6·37-s + 2.84e4·39-s − 7.37e6·41-s + 2.06e7·43-s + 3.52e7·45-s − 2.03e7·47-s + 5.76e6·49-s − 7.71e4·51-s − 5.97e7·53-s + 3.11e7·55-s + ⋯ |
| L(s) = 1 | − 0.00165·3-s − 1.28·5-s − 0.377·7-s − 0.999·9-s − 0.358·11-s − 1.18·13-s + 0.00212·15-s + 0.963·17-s − 1.34·19-s + 0.000625·21-s − 0.918·23-s + 0.643·25-s + 0.00331·27-s + 0.166·29-s + 1.04·31-s + 0.000593·33-s + 0.484·35-s − 0.266·37-s + 0.00197·39-s − 0.407·41-s + 0.923·43-s + 1.28·45-s − 0.608·47-s + 0.142·49-s − 0.00159·51-s − 1.03·53-s + 0.459·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.5103469583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5103469583\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 3 | \( 1 + 0.232T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.79e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 1.74e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.22e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.31e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 7.61e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.23e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.34e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.38e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 3.03e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 7.37e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.06e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.03e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.97e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 6.03e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.44e6T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.19e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 5.58e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.54e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.51e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.34e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.51e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.42e9T + 7.60e17T^{2} \) |
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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
−11.98589561538411938227919978693, −10.90576487845059451001259372811, −9.770164094760371469225003168242, −8.354636365588635050277368087832, −7.70196614638792401387114928651, −6.33256428097613863551267856096, −4.92475904448214918283095815291, −3.66372688289524775516944803269, −2.49515141905530250310974609915, −0.36422511955062043286326279899,
0.36422511955062043286326279899, 2.49515141905530250310974609915, 3.66372688289524775516944803269, 4.92475904448214918283095815291, 6.33256428097613863551267856096, 7.70196614638792401387114928651, 8.354636365588635050277368087832, 9.770164094760371469225003168242, 10.90576487845059451001259372811, 11.98589561538411938227919978693
