| L(s) = 1 | + 793.·2-s − 2.92e4·3-s + 1.04e5·4-s − 2.22e6·5-s − 2.31e7·6-s + 2.09e7·7-s − 3.32e8·8-s − 3.08e8·9-s − 1.76e9·10-s + 9.01e9·11-s − 3.06e9·12-s − 3.52e10·13-s + 1.66e10·14-s + 6.51e10·15-s − 3.18e11·16-s − 1.17e11·17-s − 2.44e11·18-s − 1.57e12·19-s − 2.33e11·20-s − 6.11e11·21-s + 7.14e12·22-s − 1.59e13·23-s + 9.71e12·24-s − 1.41e13·25-s − 2.79e13·26-s + 4.29e13·27-s + 2.19e12·28-s + ⋯ |
| L(s) = 1 | + 1.09·2-s − 0.856·3-s + 0.200·4-s − 0.510·5-s − 0.938·6-s + 0.196·7-s − 0.876·8-s − 0.265·9-s − 0.559·10-s + 1.15·11-s − 0.171·12-s − 0.922·13-s + 0.214·14-s + 0.437·15-s − 1.16·16-s − 0.241·17-s − 0.291·18-s − 1.12·19-s − 0.102·20-s − 0.168·21-s + 1.26·22-s − 1.84·23-s + 0.750·24-s − 0.739·25-s − 1.01·26-s + 1.08·27-s + 0.0392·28-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
| \(L(10)\) |
\(\approx\) |
\(1.031172745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.031172745\) |
| \(L(\frac{21}{2})\) |
|
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 | \( 1 - 793.T + 5.24e5T^{2} \) |
| 3 | \( 1 + 2.92e4T + 1.16e9T^{2} \) |
| 5 | \( 1 + 2.22e6T + 1.90e13T^{2} \) |
| 7 | \( 1 - 2.09e7T + 1.13e16T^{2} \) |
| 11 | \( 1 - 9.01e9T + 6.11e19T^{2} \) |
| 13 | \( 1 + 3.52e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 1.17e11T + 2.39e23T^{2} \) |
| 19 | \( 1 + 1.57e12T + 1.97e24T^{2} \) |
| 23 | \( 1 + 1.59e13T + 7.46e25T^{2} \) |
| 29 | \( 1 - 1.26e13T + 6.10e27T^{2} \) |
| 31 | \( 1 + 7.38e13T + 2.16e28T^{2} \) |
| 37 | \( 1 - 1.22e15T + 6.24e29T^{2} \) |
| 41 | \( 1 + 1.01e15T + 4.39e30T^{2} \) |
| 43 | \( 1 - 3.85e15T + 1.08e31T^{2} \) |
| 53 | \( 1 + 2.02e16T + 5.77e32T^{2} \) |
| 59 | \( 1 - 8.38e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 6.84e16T + 8.34e33T^{2} \) |
| 67 | \( 1 - 2.92e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 3.78e17T + 1.49e35T^{2} \) |
| 73 | \( 1 - 1.06e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 1.62e18T + 1.13e36T^{2} \) |
| 83 | \( 1 - 3.51e17T + 2.90e36T^{2} \) |
| 89 | \( 1 - 6.24e18T + 1.09e37T^{2} \) |
| 97 | \( 1 + 1.24e19T + 5.60e37T^{2} \) |
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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
−11.92324866075217677870419925540, −11.27190287175692675456784503714, −9.637740992208963316771019866631, −8.237144164793094673218942278588, −6.58335266917438177143398957994, −5.76679016993389105820005715711, −4.55593410347139858017877566204, −3.83102274637822412116847378341, −2.25333292498287796985170525057, −0.40827588846981197999520482813,
0.40827588846981197999520482813, 2.25333292498287796985170525057, 3.83102274637822412116847378341, 4.55593410347139858017877566204, 5.76679016993389105820005715711, 6.58335266917438177143398957994, 8.237144164793094673218942278588, 9.637740992208963316771019866631, 11.27190287175692675456784503714, 11.92324866075217677870419925540
