| L(s) = 1 | − 2-s − 1.94·3-s + 4-s + 3.69·5-s + 1.94·6-s − 2.11·7-s − 8-s + 0.782·9-s − 3.69·10-s + 1.68·11-s − 1.94·12-s + 4.38·13-s + 2.11·14-s − 7.19·15-s + 16-s + 1.07·17-s − 0.782·18-s + 0.525·19-s + 3.69·20-s + 4.10·21-s − 1.68·22-s − 0.463·23-s + 1.94·24-s + 8.67·25-s − 4.38·26-s + 4.31·27-s − 2.11·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.12·3-s + 0.5·4-s + 1.65·5-s + 0.793·6-s − 0.797·7-s − 0.353·8-s + 0.260·9-s − 1.16·10-s + 0.506·11-s − 0.561·12-s + 1.21·13-s + 0.564·14-s − 1.85·15-s + 0.250·16-s + 0.259·17-s − 0.184·18-s + 0.120·19-s + 0.826·20-s + 0.895·21-s − 0.358·22-s − 0.0967·23-s + 0.396·24-s + 1.73·25-s − 0.859·26-s + 0.830·27-s − 0.398·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 862 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9767215260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9767215260\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + 1.94T + 3T^{2} \) |
| 5 | \( 1 - 3.69T + 5T^{2} \) |
| 7 | \( 1 + 2.11T + 7T^{2} \) |
| 11 | \( 1 - 1.68T + 11T^{2} \) |
| 13 | \( 1 - 4.38T + 13T^{2} \) |
| 17 | \( 1 - 1.07T + 17T^{2} \) |
| 19 | \( 1 - 0.525T + 19T^{2} \) |
| 23 | \( 1 + 0.463T + 23T^{2} \) |
| 29 | \( 1 + 9.66T + 29T^{2} \) |
| 31 | \( 1 + 9.57T + 31T^{2} \) |
| 37 | \( 1 - 9.09T + 37T^{2} \) |
| 41 | \( 1 - 8.45T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 3.30T + 47T^{2} \) |
| 53 | \( 1 - 3.73T + 53T^{2} \) |
| 59 | \( 1 - 6.67T + 59T^{2} \) |
| 61 | \( 1 - 0.444T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 6.94T + 71T^{2} \) |
| 73 | \( 1 - 2.36T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 8.78T + 83T^{2} \) |
| 89 | \( 1 - 9.82T + 89T^{2} \) |
| 97 | \( 1 - 8.50T + 97T^{2} \) |
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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
−10.07821065773379726394383621695, −9.399244311390401173441220164486, −8.908602897429034142208258397766, −7.44970352868895002490540754243, −6.36332400333869620016754066348, −6.01050858668370635061648891260, −5.39848228313376513312921754988, −3.68029316881676265954060674277, −2.21953324977738327032463579201, −0.966330481356713775991703942349,
0.966330481356713775991703942349, 2.21953324977738327032463579201, 3.68029316881676265954060674277, 5.39848228313376513312921754988, 6.01050858668370635061648891260, 6.36332400333869620016754066348, 7.44970352868895002490540754243, 8.908602897429034142208258397766, 9.399244311390401173441220164486, 10.07821065773379726394383621695
