L(s) = 1 | − 1.69·2-s + 0.888·4-s + 0.949·5-s + 1.88·8-s − 1.61·10-s − 0.588·11-s + 5.01·13-s − 4.98·16-s + 7.58·17-s + 4.46·19-s + 0.843·20-s + 22-s + 2.47·23-s − 4.09·25-s − 8.53·26-s + 5.47·29-s + 6.07·31-s + 4.69·32-s − 12.8·34-s − 6.98·37-s − 7.58·38-s + 1.79·40-s + 1.05·41-s + 6.98·43-s − 0.522·44-s − 4.21·46-s + 7.47·47-s + ⋯ |
L(s) = 1 | − 1.20·2-s + 0.444·4-s + 0.424·5-s + 0.667·8-s − 0.510·10-s − 0.177·11-s + 1.39·13-s − 1.24·16-s + 1.83·17-s + 1.02·19-s + 0.188·20-s + 0.213·22-s + 0.516·23-s − 0.819·25-s − 1.67·26-s + 1.01·29-s + 1.09·31-s + 0.830·32-s − 2.21·34-s − 1.14·37-s − 1.23·38-s + 0.283·40-s + 0.164·41-s + 1.06·43-s − 0.0788·44-s − 0.620·46-s + 1.09·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.306305047\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.306305047\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 1.69T + 2T^{2} \) |
| 5 | \( 1 - 0.949T + 5T^{2} \) |
| 11 | \( 1 + 0.588T + 11T^{2} \) |
| 13 | \( 1 - 5.01T + 13T^{2} \) |
| 17 | \( 1 - 7.58T + 17T^{2} \) |
| 19 | \( 1 - 4.46T + 19T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 - 5.47T + 29T^{2} \) |
| 31 | \( 1 - 6.07T + 31T^{2} \) |
| 37 | \( 1 + 6.98T + 37T^{2} \) |
| 41 | \( 1 - 1.05T + 41T^{2} \) |
| 43 | \( 1 - 6.98T + 43T^{2} \) |
| 47 | \( 1 - 7.47T + 47T^{2} \) |
| 53 | \( 1 - 6.92T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 4.30T + 71T^{2} \) |
| 73 | \( 1 + 4.46T + 73T^{2} \) |
| 79 | \( 1 + 1.33T + 79T^{2} \) |
| 83 | \( 1 - 5.68T + 83T^{2} \) |
| 89 | \( 1 + 0.843T + 89T^{2} \) |
| 97 | \( 1 - 3.40T + 97T^{2} \) |
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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
−8.591641661191468602721972602572, −7.75488115841209621537732886380, −7.40451666556549899665631471273, −6.26923023266850354377242049163, −5.67099926889652792105429773052, −4.76647130340008837554442608302, −3.70405601715430039924753512822, −2.81843840527343720068408973709, −1.47193594344431941904935909953, −0.908011565394066867124676276782,
0.908011565394066867124676276782, 1.47193594344431941904935909953, 2.81843840527343720068408973709, 3.70405601715430039924753512822, 4.76647130340008837554442608302, 5.67099926889652792105429773052, 6.26923023266850354377242049163, 7.40451666556549899665631471273, 7.75488115841209621537732886380, 8.591641661191468602721972602572