L(s) = 1 | − 1.27·2-s + 2.68·3-s − 0.367·4-s − 5-s − 3.43·6-s − 7-s + 3.02·8-s + 4.23·9-s + 1.27·10-s − 0.987·12-s + 5.71·13-s + 1.27·14-s − 2.68·15-s − 3.13·16-s − 0.114·17-s − 5.40·18-s + 6.33·19-s + 0.367·20-s − 2.68·21-s − 5.42·23-s + 8.13·24-s + 25-s − 7.30·26-s + 3.31·27-s + 0.367·28-s + 3.52·29-s + 3.43·30-s + ⋯ |
L(s) = 1 | − 0.903·2-s + 1.55·3-s − 0.183·4-s − 0.447·5-s − 1.40·6-s − 0.377·7-s + 1.06·8-s + 1.41·9-s + 0.404·10-s − 0.285·12-s + 1.58·13-s + 0.341·14-s − 0.694·15-s − 0.782·16-s − 0.0277·17-s − 1.27·18-s + 1.45·19-s + 0.0821·20-s − 0.586·21-s − 1.13·23-s + 1.66·24-s + 0.200·25-s − 1.43·26-s + 0.637·27-s + 0.0694·28-s + 0.655·29-s + 0.627·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.850225406\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.850225406\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.27T + 2T^{2} \) |
| 3 | \( 1 - 2.68T + 3T^{2} \) |
| 13 | \( 1 - 5.71T + 13T^{2} \) |
| 17 | \( 1 + 0.114T + 17T^{2} \) |
| 19 | \( 1 - 6.33T + 19T^{2} \) |
| 23 | \( 1 + 5.42T + 23T^{2} \) |
| 29 | \( 1 - 3.52T + 29T^{2} \) |
| 31 | \( 1 + 7.76T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 2.05T + 41T^{2} \) |
| 43 | \( 1 - 3.25T + 43T^{2} \) |
| 47 | \( 1 + 0.743T + 47T^{2} \) |
| 53 | \( 1 - 0.656T + 53T^{2} \) |
| 59 | \( 1 + 5.88T + 59T^{2} \) |
| 61 | \( 1 - 7.98T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 + 0.829T + 71T^{2} \) |
| 73 | \( 1 + 1.17T + 73T^{2} \) |
| 79 | \( 1 + 0.699T + 79T^{2} \) |
| 83 | \( 1 + 8.74T + 83T^{2} \) |
| 89 | \( 1 - 8.44T + 89T^{2} \) |
| 97 | \( 1 + 7.68T + 97T^{2} \) |
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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
−8.523877093443906005601875195364, −7.80793296499914610153040635404, −7.51299828958621669461239276874, −6.47428229591443159983954910069, −5.44435096148927760868413275405, −4.19362077535801703880106367245, −3.74531366415385146452442066776, −2.96303546833957732912359404280, −1.82909213336936799541082374939, −0.864666209158323276601666248789,
0.864666209158323276601666248789, 1.82909213336936799541082374939, 2.96303546833957732912359404280, 3.74531366415385146452442066776, 4.19362077535801703880106367245, 5.44435096148927760868413275405, 6.47428229591443159983954910069, 7.51299828958621669461239276874, 7.80793296499914610153040635404, 8.523877093443906005601875195364