| L(s) = 1 | − 2.10·2-s − 2.66·3-s + 2.44·4-s − 3.46·5-s + 5.62·6-s − 1.09·7-s − 0.937·8-s + 4.11·9-s + 7.29·10-s − 2.77·11-s − 6.51·12-s + 4.41·13-s + 2.31·14-s + 9.22·15-s − 2.91·16-s + 17-s − 8.66·18-s + 7.84·19-s − 8.45·20-s + 2.93·21-s + 5.85·22-s − 5.90·23-s + 2.50·24-s + 6.97·25-s − 9.30·26-s − 2.96·27-s − 2.68·28-s + ⋯ |
| L(s) = 1 | − 1.49·2-s − 1.53·3-s + 1.22·4-s − 1.54·5-s + 2.29·6-s − 0.415·7-s − 0.331·8-s + 1.37·9-s + 2.30·10-s − 0.837·11-s − 1.88·12-s + 1.22·13-s + 0.619·14-s + 2.38·15-s − 0.728·16-s + 0.242·17-s − 2.04·18-s + 1.80·19-s − 1.89·20-s + 0.639·21-s + 1.24·22-s − 1.23·23-s + 0.510·24-s + 1.39·25-s − 1.82·26-s − 0.570·27-s − 0.507·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 353 | \( 1 - T \) |
| good | 2 | \( 1 + 2.10T + 2T^{2} \) |
| 3 | \( 1 + 2.66T + 3T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 + 1.09T + 7T^{2} \) |
| 11 | \( 1 + 2.77T + 11T^{2} \) |
| 13 | \( 1 - 4.41T + 13T^{2} \) |
| 19 | \( 1 - 7.84T + 19T^{2} \) |
| 23 | \( 1 + 5.90T + 23T^{2} \) |
| 29 | \( 1 + 9.04T + 29T^{2} \) |
| 31 | \( 1 + 8.72T + 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 - 5.14T + 41T^{2} \) |
| 43 | \( 1 + 4.00T + 43T^{2} \) |
| 47 | \( 1 - 9.30T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 + 7.54T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 - 1.64T + 67T^{2} \) |
| 71 | \( 1 - 0.983T + 71T^{2} \) |
| 73 | \( 1 - 8.86T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 - 16.2T + 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 2.51T + 97T^{2} \) |
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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
−7.72537282898875053388922879016, −7.28571865232256356903797154145, −6.54324215801028742837227376120, −5.64293129246260148997566660649, −5.08997108798718320221153092091, −3.93302250182749500889860575159, −3.37843891054111357983170221248, −1.73129989382973694288708957492, −0.68207559884068156493257976183, 0,
0.68207559884068156493257976183, 1.73129989382973694288708957492, 3.37843891054111357983170221248, 3.93302250182749500889860575159, 5.08997108798718320221153092091, 5.64293129246260148997566660649, 6.54324215801028742837227376120, 7.28571865232256356903797154145, 7.72537282898875053388922879016
