| L(s) = 1 | + 2.17·2-s + 3-s + 2.70·4-s − 0.539·5-s + 2.17·6-s + 0.630·7-s + 1.53·8-s + 9-s − 1.17·10-s + 11-s + 2.70·12-s + 13-s + 1.36·14-s − 0.539·15-s − 2.07·16-s + 1.90·17-s + 2.17·18-s − 4.04·19-s − 1.46·20-s + 0.630·21-s + 2.17·22-s + 1.36·23-s + 1.53·24-s − 4.70·25-s + 2.17·26-s + 27-s + 1.70·28-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 0.577·3-s + 1.35·4-s − 0.241·5-s + 0.885·6-s + 0.238·7-s + 0.544·8-s + 0.333·9-s − 0.370·10-s + 0.301·11-s + 0.782·12-s + 0.277·13-s + 0.365·14-s − 0.139·15-s − 0.519·16-s + 0.462·17-s + 0.511·18-s − 0.929·19-s − 0.326·20-s + 0.137·21-s + 0.462·22-s + 0.285·23-s + 0.314·24-s − 0.941·25-s + 0.425·26-s + 0.192·27-s + 0.323·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.481566823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.481566823\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 5 | \( 1 + 0.539T + 5T^{2} \) |
| 7 | \( 1 - 0.630T + 7T^{2} \) |
| 17 | \( 1 - 1.90T + 17T^{2} \) |
| 19 | \( 1 + 4.04T + 19T^{2} \) |
| 23 | \( 1 - 1.36T + 23T^{2} \) |
| 29 | \( 1 + 2.24T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 + 5.07T + 37T^{2} \) |
| 41 | \( 1 + 2.04T + 41T^{2} \) |
| 43 | \( 1 - 0.986T + 43T^{2} \) |
| 47 | \( 1 - 3.26T + 47T^{2} \) |
| 53 | \( 1 - 9.91T + 53T^{2} \) |
| 59 | \( 1 + 3.60T + 59T^{2} \) |
| 61 | \( 1 - 3.41T + 61T^{2} \) |
| 67 | \( 1 - 5.95T + 67T^{2} \) |
| 71 | \( 1 + 7.75T + 71T^{2} \) |
| 73 | \( 1 - 9.46T + 73T^{2} \) |
| 79 | \( 1 - 3.90T + 79T^{2} \) |
| 83 | \( 1 - 2.34T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + 5.07T + 97T^{2} \) |
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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
−11.49006928712101299019888361723, −10.53479025824354864820628599331, −9.292401800608415495554552223062, −8.324844799775443177843258341885, −7.23793320569824366802772753851, −6.25178769815951542025593896712, −5.22267141746468630029700130120, −4.14166044322649030792283551854, −3.41808635831160283557764191395, −2.07793111841086217289691063132,
2.07793111841086217289691063132, 3.41808635831160283557764191395, 4.14166044322649030792283551854, 5.22267141746468630029700130120, 6.25178769815951542025593896712, 7.23793320569824366802772753851, 8.324844799775443177843258341885, 9.292401800608415495554552223062, 10.53479025824354864820628599331, 11.49006928712101299019888361723
