| L(s) = 1 | − 2.04·2-s − 3-s + 2.18·4-s − 2.05·5-s + 2.04·6-s − 7-s − 0.385·8-s + 9-s + 4.20·10-s + 0.528·11-s − 2.18·12-s − 2.96·13-s + 2.04·14-s + 2.05·15-s − 3.58·16-s − 5.27·17-s − 2.04·18-s − 4.01·19-s − 4.49·20-s + 21-s − 1.08·22-s + 7.89·23-s + 0.385·24-s − 0.772·25-s + 6.06·26-s − 27-s − 2.18·28-s + ⋯ |
| L(s) = 1 | − 1.44·2-s − 0.577·3-s + 1.09·4-s − 0.919·5-s + 0.835·6-s − 0.377·7-s − 0.136·8-s + 0.333·9-s + 1.33·10-s + 0.159·11-s − 0.631·12-s − 0.821·13-s + 0.546·14-s + 0.530·15-s − 0.896·16-s − 1.28·17-s − 0.482·18-s − 0.920·19-s − 1.00·20-s + 0.218·21-s − 0.230·22-s + 1.64·23-s + 0.0787·24-s − 0.154·25-s + 1.18·26-s − 0.192·27-s − 0.413·28-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
| \(L(1)\) |
\(\approx\) |
\(0.1847738540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1847738540\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.04T + 2T^{2} \) |
| 5 | \( 1 + 2.05T + 5T^{2} \) |
| 11 | \( 1 - 0.528T + 11T^{2} \) |
| 13 | \( 1 + 2.96T + 13T^{2} \) |
| 17 | \( 1 + 5.27T + 17T^{2} \) |
| 19 | \( 1 + 4.01T + 19T^{2} \) |
| 23 | \( 1 - 7.89T + 23T^{2} \) |
| 29 | \( 1 - 1.95T + 29T^{2} \) |
| 31 | \( 1 + 4.78T + 31T^{2} \) |
| 37 | \( 1 + 1.17T + 37T^{2} \) |
| 41 | \( 1 + 4.96T + 41T^{2} \) |
| 43 | \( 1 - 3.63T + 43T^{2} \) |
| 47 | \( 1 - 2.64T + 47T^{2} \) |
| 53 | \( 1 + 2.95T + 53T^{2} \) |
| 59 | \( 1 + 1.25T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 0.487T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 + 7.25T + 79T^{2} \) |
| 83 | \( 1 + 5.81T + 83T^{2} \) |
| 89 | \( 1 + 7.38T + 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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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
−8.934549008511456466963543223528, −8.219200790270107019026982655087, −7.21277026102885864606321811750, −7.06549581281936499169455689075, −6.07805809871324369382759970614, −4.78984235753684344039614902175, −4.22920541626436148555061065031, −2.89634494281901919655423678217, −1.71588949444512897369433918986, −0.33891506198028335854019314987,
0.33891506198028335854019314987, 1.71588949444512897369433918986, 2.89634494281901919655423678217, 4.22920541626436148555061065031, 4.78984235753684344039614902175, 6.07805809871324369382759970614, 7.06549581281936499169455689075, 7.21277026102885864606321811750, 8.219200790270107019026982655087, 8.934549008511456466963543223528
