| L(s) = 1 | − 1.08e3·2-s − 3.95e4·3-s + 6.43e5·4-s + 8.62e6·5-s + 4.27e7·6-s − 1.13e8·7-s − 1.28e8·8-s + 4.05e8·9-s − 9.32e9·10-s − 7.29e9·11-s − 2.54e10·12-s + 5.60e10·13-s + 1.22e11·14-s − 3.41e11·15-s − 1.98e11·16-s − 8.31e11·17-s − 4.37e11·18-s + 2.22e12·19-s + 5.54e12·20-s + 4.47e12·21-s + 7.88e12·22-s + 6.38e12·23-s + 5.08e12·24-s + 5.53e13·25-s − 6.05e13·26-s + 2.99e13·27-s − 7.27e13·28-s + ⋯ |
| L(s) = 1 | − 1.49·2-s − 1.16·3-s + 1.22·4-s + 1.97·5-s + 1.73·6-s − 1.05·7-s − 0.338·8-s + 0.348·9-s − 2.94·10-s − 0.933·11-s − 1.42·12-s + 1.46·13-s + 1.58·14-s − 2.29·15-s − 0.722·16-s − 1.70·17-s − 0.520·18-s + 1.58·19-s + 2.42·20-s + 1.23·21-s + 1.39·22-s + 0.739·23-s + 0.392·24-s + 2.90·25-s − 2.18·26-s + 0.756·27-s − 1.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(0.6813140061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6813140061\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 47 | \( 1 + 1.11e15T \) |
| good | 2 | \( 1 + 1.08e3T + 5.24e5T^{2} \) |
| 3 | \( 1 + 3.95e4T + 1.16e9T^{2} \) |
| 5 | \( 1 - 8.62e6T + 1.90e13T^{2} \) |
| 7 | \( 1 + 1.13e8T + 1.13e16T^{2} \) |
| 11 | \( 1 + 7.29e9T + 6.11e19T^{2} \) |
| 13 | \( 1 - 5.60e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 8.31e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 2.22e12T + 1.97e24T^{2} \) |
| 23 | \( 1 - 6.38e12T + 7.46e25T^{2} \) |
| 29 | \( 1 + 5.23e13T + 6.10e27T^{2} \) |
| 31 | \( 1 + 7.78e13T + 2.16e28T^{2} \) |
| 37 | \( 1 + 2.80e14T + 6.24e29T^{2} \) |
| 41 | \( 1 - 2.10e15T + 4.39e30T^{2} \) |
| 43 | \( 1 + 1.60e15T + 1.08e31T^{2} \) |
| 53 | \( 1 - 1.90e16T + 5.77e32T^{2} \) |
| 59 | \( 1 + 6.49e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 1.11e17T + 8.34e33T^{2} \) |
| 67 | \( 1 + 3.69e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 3.81e17T + 1.49e35T^{2} \) |
| 73 | \( 1 - 7.52e15T + 2.53e35T^{2} \) |
| 79 | \( 1 - 6.63e17T + 1.13e36T^{2} \) |
| 83 | \( 1 + 1.09e18T + 2.90e36T^{2} \) |
| 89 | \( 1 + 2.45e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 1.12e19T + 5.60e37T^{2} \) |
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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
−11.11750827492673688592307155387, −10.54330385648599811813331072224, −9.545275149993699395517843810517, −8.853998686776038062923900002401, −6.91064820723280922951205017833, −6.14149740546034236785657053277, −5.25131559098509836629545805532, −2.73312994426189358778968707557, −1.51094728931534212758622356510, −0.54940273614613963581693823486,
0.54940273614613963581693823486, 1.51094728931534212758622356510, 2.73312994426189358778968707557, 5.25131559098509836629545805532, 6.14149740546034236785657053277, 6.91064820723280922951205017833, 8.853998686776038062923900002401, 9.545275149993699395517843810517, 10.54330385648599811813331072224, 11.11750827492673688592307155387
