| L(s) = 1 | + 3.00e6·3-s + 9.02e8·5-s − 3.76e11·7-s + 1.42e12·9-s + 1.19e14·11-s + 8.45e14·13-s + 2.71e15·15-s − 5.80e16·17-s + 2.28e16·19-s − 1.13e18·21-s + 4.11e18·23-s − 6.63e18·25-s − 1.86e19·27-s − 5.09e19·29-s − 2.28e20·31-s + 3.60e20·33-s − 3.40e20·35-s − 1.37e21·37-s + 2.54e21·39-s + 1.26e21·41-s + 1.57e22·43-s + 1.28e21·45-s − 1.74e22·47-s + 7.64e22·49-s − 1.74e23·51-s − 2.42e23·53-s + 1.08e23·55-s + ⋯ |
| L(s) = 1 | + 1.08·3-s + 0.330·5-s − 1.47·7-s + 0.186·9-s + 1.04·11-s + 0.774·13-s + 0.360·15-s − 1.42·17-s + 0.124·19-s − 1.60·21-s + 1.70·23-s − 0.890·25-s − 0.885·27-s − 0.921·29-s − 1.68·31-s + 1.14·33-s − 0.486·35-s − 0.925·37-s + 0.843·39-s + 0.212·41-s + 1.40·43-s + 0.0617·45-s − 0.465·47-s + 1.16·49-s − 1.54·51-s − 1.28·53-s + 0.346·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(14)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{29}{2})\) |
|
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 | \( 1 - 3.00e6T + 7.62e12T^{2} \) |
| 5 | \( 1 - 9.02e8T + 7.45e18T^{2} \) |
| 7 | \( 1 + 3.76e11T + 6.57e22T^{2} \) |
| 11 | \( 1 - 1.19e14T + 1.31e28T^{2} \) |
| 13 | \( 1 - 8.45e14T + 1.19e30T^{2} \) |
| 17 | \( 1 + 5.80e16T + 1.66e33T^{2} \) |
| 19 | \( 1 - 2.28e16T + 3.36e34T^{2} \) |
| 23 | \( 1 - 4.11e18T + 5.84e36T^{2} \) |
| 29 | \( 1 + 5.09e19T + 3.05e39T^{2} \) |
| 31 | \( 1 + 2.28e20T + 1.84e40T^{2} \) |
| 37 | \( 1 + 1.37e21T + 2.19e42T^{2} \) |
| 41 | \( 1 - 1.26e21T + 3.50e43T^{2} \) |
| 43 | \( 1 - 1.57e22T + 1.26e44T^{2} \) |
| 47 | \( 1 + 1.74e22T + 1.40e45T^{2} \) |
| 53 | \( 1 + 2.42e23T + 3.59e46T^{2} \) |
| 59 | \( 1 - 4.33e23T + 6.50e47T^{2} \) |
| 61 | \( 1 + 1.61e24T + 1.59e48T^{2} \) |
| 67 | \( 1 + 3.04e24T + 2.01e49T^{2} \) |
| 71 | \( 1 - 1.33e24T + 9.63e49T^{2} \) |
| 73 | \( 1 + 2.26e25T + 2.04e50T^{2} \) |
| 79 | \( 1 + 2.50e25T + 1.72e51T^{2} \) |
| 83 | \( 1 + 2.43e25T + 6.53e51T^{2} \) |
| 89 | \( 1 - 3.08e26T + 4.30e52T^{2} \) |
| 97 | \( 1 + 2.50e26T + 4.39e53T^{2} \) |
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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
−12.90125399241009486083222237024, −11.08038271972932137103011784095, −9.299083432650613057169076553401, −8.997511905813082241747603891968, −7.12026981783871185705887621263, −5.99495918612327610638633970875, −3.88570679795934219877029046643, −3.02663056496524052145031640765, −1.71502809173221752353720406658, 0,
1.71502809173221752353720406658, 3.02663056496524052145031640765, 3.88570679795934219877029046643, 5.99495918612327610638633970875, 7.12026981783871185705887621263, 8.997511905813082241747603891968, 9.299083432650613057169076553401, 11.08038271972932137103011784095, 12.90125399241009486083222237024
