| L(s) = 1 | + 27.3·2-s + 101.·3-s + 234.·4-s + 2.38e3·5-s + 2.76e3·6-s + 3.03e3·7-s − 7.59e3·8-s − 9.44e3·9-s + 6.51e4·10-s − 306.·11-s + 2.36e4·12-s + 1.22e5·13-s + 8.28e4·14-s + 2.41e5·15-s − 3.27e5·16-s − 2.21e5·17-s − 2.57e5·18-s + 1.01e6·19-s + 5.58e5·20-s + 3.06e5·21-s − 8.35e3·22-s − 2.08e6·23-s − 7.68e5·24-s + 3.72e6·25-s + 3.35e6·26-s − 2.94e6·27-s + 7.09e5·28-s + ⋯ |
| L(s) = 1 | + 1.20·2-s + 0.721·3-s + 0.457·4-s + 1.70·5-s + 0.870·6-s + 0.477·7-s − 0.655·8-s − 0.479·9-s + 2.05·10-s − 0.00630·11-s + 0.329·12-s + 1.19·13-s + 0.576·14-s + 1.23·15-s − 1.24·16-s − 0.643·17-s − 0.579·18-s + 1.78·19-s + 0.779·20-s + 0.344·21-s − 0.00760·22-s − 1.55·23-s − 0.472·24-s + 1.90·25-s + 1.43·26-s − 1.06·27-s + 0.218·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.988051149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.988051149\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 7.07e5T \) |
| good | 2 | \( 1 - 27.3T + 512T^{2} \) |
| 3 | \( 1 - 101.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.38e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 3.03e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 306.T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.22e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.21e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.01e6T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.08e6T + 1.80e12T^{2} \) |
| 31 | \( 1 + 7.08e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 8.45e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.07e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 8.62e5T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.02e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.71e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.38e6T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.88e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.69e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.13e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.89e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 9.46e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.99e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.49e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 2.76e8T + 7.60e17T^{2} \) |
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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
−14.33627871917935858099810400043, −13.91258958760882741762382836310, −13.19351480150019374998262059962, −11.52393786758618593838593881326, −9.729780167162305953743282164172, −8.570858951233030629123907533503, −6.22639877693991319231203051821, −5.23679666449663170926261908649, −3.33044649825447512868367019653, −1.90337319429377550959650635497,
1.90337319429377550959650635497, 3.33044649825447512868367019653, 5.23679666449663170926261908649, 6.22639877693991319231203051821, 8.570858951233030629123907533503, 9.729780167162305953743282164172, 11.52393786758618593838593881326, 13.19351480150019374998262059962, 13.91258958760882741762382836310, 14.33627871917935858099810400043
