| L(s) = 1 | + 1.09·2-s − 3-s − 0.804·4-s + 1.42·5-s − 1.09·6-s − 7-s − 3.06·8-s + 9-s + 1.55·10-s − 4.17·11-s + 0.804·12-s + 4.20·13-s − 1.09·14-s − 1.42·15-s − 1.74·16-s + 5.66·17-s + 1.09·18-s + 1.69·19-s − 1.14·20-s + 21-s − 4.56·22-s − 0.943·23-s + 3.06·24-s − 2.97·25-s + 4.59·26-s − 27-s + 0.804·28-s + ⋯ |
| L(s) = 1 | + 0.773·2-s − 0.577·3-s − 0.402·4-s + 0.636·5-s − 0.446·6-s − 0.377·7-s − 1.08·8-s + 0.333·9-s + 0.491·10-s − 1.25·11-s + 0.232·12-s + 1.16·13-s − 0.292·14-s − 0.367·15-s − 0.436·16-s + 1.37·17-s + 0.257·18-s + 0.389·19-s − 0.255·20-s + 0.218·21-s − 0.973·22-s − 0.196·23-s + 0.625·24-s − 0.595·25-s + 0.901·26-s − 0.192·27-s + 0.151·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.771457864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.771457864\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 1.09T + 2T^{2} \) |
| 5 | \( 1 - 1.42T + 5T^{2} \) |
| 11 | \( 1 + 4.17T + 11T^{2} \) |
| 13 | \( 1 - 4.20T + 13T^{2} \) |
| 17 | \( 1 - 5.66T + 17T^{2} \) |
| 19 | \( 1 - 1.69T + 19T^{2} \) |
| 23 | \( 1 + 0.943T + 23T^{2} \) |
| 29 | \( 1 + 7.23T + 29T^{2} \) |
| 31 | \( 1 + 3.64T + 31T^{2} \) |
| 37 | \( 1 + 3.63T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 5.58T + 43T^{2} \) |
| 47 | \( 1 - 8.66T + 47T^{2} \) |
| 53 | \( 1 - 6.51T + 53T^{2} \) |
| 59 | \( 1 - 3.57T + 59T^{2} \) |
| 61 | \( 1 - 7.68T + 61T^{2} \) |
| 67 | \( 1 - 7.65T + 67T^{2} \) |
| 71 | \( 1 + 6.76T + 71T^{2} \) |
| 73 | \( 1 + 5.73T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 + 0.799T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 + 4.78T + 97T^{2} \) |
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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
−9.023697863219040026177899224539, −7.987787631170951338834577786278, −7.27174052369326414709041686498, −6.00341895260300120287662930905, −5.70713101011616907780533016423, −5.24241926039807102068230837833, −4.02709282693416659379778492500, −3.42316744316547557838073590112, −2.27843833669014825243969308217, −0.76268787514968296276467914431,
0.76268787514968296276467914431, 2.27843833669014825243969308217, 3.42316744316547557838073590112, 4.02709282693416659379778492500, 5.24241926039807102068230837833, 5.70713101011616907780533016423, 6.00341895260300120287662930905, 7.27174052369326414709041686498, 7.987787631170951338834577786278, 9.023697863219040026177899224539
