| L(s) = 1 | − 2.61·2-s − 2.31·3-s + 4.86·4-s + 0.216·5-s + 6.07·6-s + 3.38·7-s − 7.50·8-s + 2.37·9-s − 0.566·10-s + 0.605·11-s − 11.2·12-s − 3.04·13-s − 8.86·14-s − 0.501·15-s + 9.92·16-s + 17-s − 6.22·18-s + 4.98·19-s + 1.05·20-s − 7.84·21-s − 1.58·22-s + 1.60·23-s + 17.3·24-s − 4.95·25-s + 7.98·26-s + 1.44·27-s + 16.4·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 1.33·3-s + 2.43·4-s + 0.0966·5-s + 2.47·6-s + 1.27·7-s − 2.65·8-s + 0.792·9-s − 0.179·10-s + 0.182·11-s − 3.25·12-s − 0.845·13-s − 2.36·14-s − 0.129·15-s + 2.48·16-s + 0.242·17-s − 1.46·18-s + 1.14·19-s + 0.235·20-s − 1.71·21-s − 0.338·22-s + 0.334·23-s + 3.55·24-s − 0.990·25-s + 1.56·26-s + 0.278·27-s + 3.11·28-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 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 + 2.31T + 3T^{2} \) |
| 5 | \( 1 - 0.216T + 5T^{2} \) |
| 7 | \( 1 - 3.38T + 7T^{2} \) |
| 11 | \( 1 - 0.605T + 11T^{2} \) |
| 13 | \( 1 + 3.04T + 13T^{2} \) |
| 19 | \( 1 - 4.98T + 19T^{2} \) |
| 23 | \( 1 - 1.60T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 - 5.87T + 31T^{2} \) |
| 37 | \( 1 + 3.75T + 37T^{2} \) |
| 41 | \( 1 + 0.864T + 41T^{2} \) |
| 43 | \( 1 - 6.40T + 43T^{2} \) |
| 47 | \( 1 - 0.621T + 47T^{2} \) |
| 53 | \( 1 - 9.64T + 53T^{2} \) |
| 59 | \( 1 + 5.81T + 59T^{2} \) |
| 61 | \( 1 + 6.34T + 61T^{2} \) |
| 67 | \( 1 - 8.53T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 16.5T + 73T^{2} \) |
| 79 | \( 1 - 0.298T + 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 + 6.23T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
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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
−7.78136939601586060238016967274, −7.22231493571902211913696767386, −6.58688710731055504079855737781, −5.52906031061119277854066364553, −5.35675093660156124404044123649, −4.16364432965896059725884162212, −2.73960607107949825331055199417, −1.75281251979335595503666476437, −1.05788110457145036196710833418, 0,
1.05788110457145036196710833418, 1.75281251979335595503666476437, 2.73960607107949825331055199417, 4.16364432965896059725884162212, 5.35675093660156124404044123649, 5.52906031061119277854066364553, 6.58688710731055504079855737781, 7.22231493571902211913696767386, 7.78136939601586060238016967274
