| L(s) = 1 | − 2.75·2-s − 1.38·3-s + 5.58·4-s + 1.31·5-s + 3.81·6-s − 4.81·7-s − 9.88·8-s − 1.08·9-s − 3.62·10-s − 5.09·11-s − 7.73·12-s + 5.27·13-s + 13.2·14-s − 1.82·15-s + 16.0·16-s + 17-s + 2.98·18-s + 6.13·19-s + 7.35·20-s + 6.66·21-s + 14.0·22-s − 3.61·23-s + 13.6·24-s − 3.26·25-s − 14.5·26-s + 5.65·27-s − 26.9·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s − 0.799·3-s + 2.79·4-s + 0.588·5-s + 1.55·6-s − 1.82·7-s − 3.49·8-s − 0.361·9-s − 1.14·10-s − 1.53·11-s − 2.23·12-s + 1.46·13-s + 3.54·14-s − 0.470·15-s + 4.01·16-s + 0.242·17-s + 0.704·18-s + 1.40·19-s + 1.64·20-s + 1.45·21-s + 2.99·22-s − 0.753·23-s + 2.79·24-s − 0.653·25-s − 2.84·26-s + 1.08·27-s − 5.08·28-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 | \( 1 + 2.75T + 2T^{2} \) |
| 3 | \( 1 + 1.38T + 3T^{2} \) |
| 5 | \( 1 - 1.31T + 5T^{2} \) |
| 7 | \( 1 + 4.81T + 7T^{2} \) |
| 11 | \( 1 + 5.09T + 11T^{2} \) |
| 13 | \( 1 - 5.27T + 13T^{2} \) |
| 19 | \( 1 - 6.13T + 19T^{2} \) |
| 23 | \( 1 + 3.61T + 23T^{2} \) |
| 29 | \( 1 + 6.73T + 29T^{2} \) |
| 31 | \( 1 + 6.43T + 31T^{2} \) |
| 37 | \( 1 - 5.38T + 37T^{2} \) |
| 41 | \( 1 + 3.59T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 4.09T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 1.62T + 61T^{2} \) |
| 67 | \( 1 + 8.49T + 67T^{2} \) |
| 71 | \( 1 - 5.46T + 71T^{2} \) |
| 73 | \( 1 + 4.51T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 0.164T + 83T^{2} \) |
| 89 | \( 1 - 3.49T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−7.62877042728878232110680663048, −7.30379500189816341782348895633, −6.26011305899062087920466641802, −5.76811801910660889650846699276, −5.68905933726422786094314344745, −3.53206160331585459885453661041, −2.95249919779602115726326671598, −2.05766217891759633583347636723, −0.809129810494469397817411812974, 0,
0.809129810494469397817411812974, 2.05766217891759633583347636723, 2.95249919779602115726326671598, 3.53206160331585459885453661041, 5.68905933726422786094314344745, 5.76811801910660889650846699276, 6.26011305899062087920466641802, 7.30379500189816341782348895633, 7.62877042728878232110680663048
