| L(s) = 1 | − 1.47·2-s + 2.93·3-s + 0.181·4-s + 5-s − 4.33·6-s + 2.68·8-s + 5.60·9-s − 1.47·10-s − 11-s + 0.532·12-s − 3.85·13-s + 2.93·15-s − 4.32·16-s + 5.44·17-s − 8.27·18-s + 7.63·19-s + 0.181·20-s + 1.47·22-s − 0.920·23-s + 7.87·24-s + 25-s + 5.69·26-s + 7.62·27-s − 4.04·29-s − 4.33·30-s − 6.90·31-s + 1.02·32-s + ⋯ |
| L(s) = 1 | − 1.04·2-s + 1.69·3-s + 0.0907·4-s + 0.447·5-s − 1.76·6-s + 0.949·8-s + 1.86·9-s − 0.467·10-s − 0.301·11-s + 0.153·12-s − 1.06·13-s + 0.757·15-s − 1.08·16-s + 1.32·17-s − 1.94·18-s + 1.75·19-s + 0.0405·20-s + 0.314·22-s − 0.191·23-s + 1.60·24-s + 0.200·25-s + 1.11·26-s + 1.46·27-s − 0.750·29-s − 0.790·30-s − 1.23·31-s + 0.180·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.080130290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.080130290\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 1.47T + 2T^{2} \) |
| 3 | \( 1 - 2.93T + 3T^{2} \) |
| 13 | \( 1 + 3.85T + 13T^{2} \) |
| 17 | \( 1 - 5.44T + 17T^{2} \) |
| 19 | \( 1 - 7.63T + 19T^{2} \) |
| 23 | \( 1 + 0.920T + 23T^{2} \) |
| 29 | \( 1 + 4.04T + 29T^{2} \) |
| 31 | \( 1 + 6.90T + 31T^{2} \) |
| 37 | \( 1 - 8.92T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 7.66T + 43T^{2} \) |
| 47 | \( 1 + 7.25T + 47T^{2} \) |
| 53 | \( 1 - 1.55T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 - 8.45T + 61T^{2} \) |
| 67 | \( 1 + 2.23T + 67T^{2} \) |
| 71 | \( 1 - 8.70T + 71T^{2} \) |
| 73 | \( 1 + 2.53T + 73T^{2} \) |
| 79 | \( 1 - 5.04T + 79T^{2} \) |
| 83 | \( 1 + 8.75T + 83T^{2} \) |
| 89 | \( 1 - 6.26T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−9.091531735791065688628528060620, −7.984139099499313170933893898428, −7.62601767494764594173508614274, −7.28724041689520878497379452960, −5.73728651003996340667126812297, −4.84106765312327310219791436569, −3.79035125584880696094999031055, −2.90026275317314994007475180351, −2.06752953178101548929863921497, −1.03601828414862436190185163536,
1.03601828414862436190185163536, 2.06752953178101548929863921497, 2.90026275317314994007475180351, 3.79035125584880696094999031055, 4.84106765312327310219791436569, 5.73728651003996340667126812297, 7.28724041689520878497379452960, 7.62601767494764594173508614274, 7.984139099499313170933893898428, 9.091531735791065688628528060620
