| L(s) = 1 | − 0.442·2-s − 3-s − 1.80·4-s + 4.29·5-s + 0.442·6-s + 1.68·8-s + 9-s − 1.89·10-s − 2.21·11-s + 1.80·12-s + 3.11·13-s − 4.29·15-s + 2.86·16-s + 6.92·17-s − 0.442·18-s − 7.72·19-s − 7.74·20-s + 0.978·22-s + 0.0381·23-s − 1.68·24-s + 13.4·25-s − 1.37·26-s − 27-s + 2.66·29-s + 1.89·30-s − 4.69·31-s − 4.63·32-s + ⋯ |
| L(s) = 1 | − 0.312·2-s − 0.577·3-s − 0.902·4-s + 1.91·5-s + 0.180·6-s + 0.594·8-s + 0.333·9-s − 0.600·10-s − 0.666·11-s + 0.520·12-s + 0.864·13-s − 1.10·15-s + 0.716·16-s + 1.67·17-s − 0.104·18-s − 1.77·19-s − 1.73·20-s + 0.208·22-s + 0.00795·23-s − 0.343·24-s + 2.68·25-s − 0.270·26-s − 0.192·27-s + 0.495·29-s + 0.346·30-s − 0.844·31-s − 0.818·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.724144084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.724144084\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 0.442T + 2T^{2} \) |
| 5 | \( 1 - 4.29T + 5T^{2} \) |
| 11 | \( 1 + 2.21T + 11T^{2} \) |
| 13 | \( 1 - 3.11T + 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 + 7.72T + 19T^{2} \) |
| 23 | \( 1 - 0.0381T + 23T^{2} \) |
| 29 | \( 1 - 2.66T + 29T^{2} \) |
| 31 | \( 1 + 4.69T + 31T^{2} \) |
| 37 | \( 1 + 0.0300T + 37T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 3.78T + 47T^{2} \) |
| 53 | \( 1 + 0.546T + 53T^{2} \) |
| 59 | \( 1 - 9.55T + 59T^{2} \) |
| 61 | \( 1 + 6.02T + 61T^{2} \) |
| 67 | \( 1 + 8.97T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 - 2.42T + 73T^{2} \) |
| 79 | \( 1 - 2.16T + 79T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 - 8.69T + 89T^{2} \) |
| 97 | \( 1 + 0.216T + 97T^{2} \) |
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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
−8.257527713228013453121491960159, −7.38741500985273718451722447524, −6.40192444584184243070056632258, −5.79755154323272388738804330782, −5.44578195265401575789783465060, −4.66057823687355770886308164236, −3.73711913515387090906461033770, −2.57620190180669231618768535728, −1.63011986823967857525372928985, −0.803039473592025986543055428113,
0.803039473592025986543055428113, 1.63011986823967857525372928985, 2.57620190180669231618768535728, 3.73711913515387090906461033770, 4.66057823687355770886308164236, 5.44578195265401575789783465060, 5.79755154323272388738804330782, 6.40192444584184243070056632258, 7.38741500985273718451722447524, 8.257527713228013453121491960159
