L(s) = 1 | − 2.02·2-s − 3-s + 2.08·4-s − 5-s + 2.02·6-s + 0.941·7-s − 0.169·8-s + 9-s + 2.02·10-s + 0.506·11-s − 2.08·12-s − 1.23·13-s − 1.90·14-s + 15-s − 3.82·16-s − 5.08·17-s − 2.02·18-s − 8.24·19-s − 2.08·20-s − 0.941·21-s − 1.02·22-s + 1.63·23-s + 0.169·24-s + 25-s + 2.48·26-s − 27-s + 1.96·28-s + ⋯ |
L(s) = 1 | − 1.42·2-s − 0.577·3-s + 1.04·4-s − 0.447·5-s + 0.825·6-s + 0.355·7-s − 0.0599·8-s + 0.333·9-s + 0.639·10-s + 0.152·11-s − 0.601·12-s − 0.341·13-s − 0.508·14-s + 0.258·15-s − 0.956·16-s − 1.23·17-s − 0.476·18-s − 1.89·19-s − 0.465·20-s − 0.205·21-s − 0.218·22-s + 0.340·23-s + 0.0345·24-s + 0.200·25-s + 0.488·26-s − 0.192·27-s + 0.370·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2445793748\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2445793748\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 + 2.02T + 2T^{2} \) |
| 7 | \( 1 - 0.941T + 7T^{2} \) |
| 11 | \( 1 - 0.506T + 11T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 + 5.08T + 17T^{2} \) |
| 19 | \( 1 + 8.24T + 19T^{2} \) |
| 23 | \( 1 - 1.63T + 23T^{2} \) |
| 29 | \( 1 + 2.04T + 29T^{2} \) |
| 31 | \( 1 - 0.257T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 41 | \( 1 + 6.66T + 41T^{2} \) |
| 43 | \( 1 - 4.47T + 43T^{2} \) |
| 47 | \( 1 + 2.85T + 47T^{2} \) |
| 53 | \( 1 + 5.20T + 53T^{2} \) |
| 59 | \( 1 - 6.60T + 59T^{2} \) |
| 61 | \( 1 + 8.27T + 61T^{2} \) |
| 67 | \( 1 + 3.93T + 67T^{2} \) |
| 71 | \( 1 - 4.15T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 7.17T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 5.21T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−8.175639710829117546099098346482, −7.54608327464397800797597203386, −6.64157222848905390877783240658, −6.49590190285675515246643516778, −5.08279190190097022942449843353, −4.57823548063547091118351238430, −3.69769172533923758609915566057, −2.27884964223730759768865801007, −1.64403375932361557332614838592, −0.33081033877093474196028663033,
0.33081033877093474196028663033, 1.64403375932361557332614838592, 2.27884964223730759768865801007, 3.69769172533923758609915566057, 4.57823548063547091118351238430, 5.08279190190097022942449843353, 6.49590190285675515246643516778, 6.64157222848905390877783240658, 7.54608327464397800797597203386, 8.175639710829117546099098346482