| L(s) = 1 | − 2.44·2-s − 2.39·3-s + 3.99·4-s − 0.961·5-s + 5.85·6-s − 7-s − 4.88·8-s + 2.71·9-s + 2.35·10-s − 6.49·11-s − 9.55·12-s − 6.00·13-s + 2.44·14-s + 2.29·15-s + 3.97·16-s − 1.46·17-s − 6.64·18-s − 4.74·19-s − 3.84·20-s + 2.39·21-s + 15.8·22-s − 0.0229·23-s + 11.6·24-s − 4.07·25-s + 14.6·26-s + 0.679·27-s − 3.99·28-s + ⋯ |
| L(s) = 1 | − 1.73·2-s − 1.38·3-s + 1.99·4-s − 0.429·5-s + 2.38·6-s − 0.377·7-s − 1.72·8-s + 0.905·9-s + 0.744·10-s − 1.95·11-s − 2.75·12-s − 1.66·13-s + 0.654·14-s + 0.593·15-s + 0.993·16-s − 0.356·17-s − 1.56·18-s − 1.08·19-s − 0.858·20-s + 0.521·21-s + 3.38·22-s − 0.00477·23-s + 2.38·24-s − 0.815·25-s + 2.88·26-s + 0.130·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + T \) |
| 863 | \( 1 + T \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 3 | \( 1 + 2.39T + 3T^{2} \) |
| 5 | \( 1 + 0.961T + 5T^{2} \) |
| 11 | \( 1 + 6.49T + 11T^{2} \) |
| 13 | \( 1 + 6.00T + 13T^{2} \) |
| 17 | \( 1 + 1.46T + 17T^{2} \) |
| 19 | \( 1 + 4.74T + 19T^{2} \) |
| 23 | \( 1 + 0.0229T + 23T^{2} \) |
| 29 | \( 1 - 1.53T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 - 3.28T + 37T^{2} \) |
| 41 | \( 1 - 5.38T + 41T^{2} \) |
| 43 | \( 1 - 6.72T + 43T^{2} \) |
| 47 | \( 1 - 0.369T + 47T^{2} \) |
| 53 | \( 1 + 1.08T + 53T^{2} \) |
| 59 | \( 1 - 5.19T + 59T^{2} \) |
| 61 | \( 1 - 7.25T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 - 3.74T + 71T^{2} \) |
| 73 | \( 1 - 4.89T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 0.272T + 83T^{2} \) |
| 89 | \( 1 - 2.33T + 89T^{2} \) |
| 97 | \( 1 - 1.22T + 97T^{2} \) |
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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
−7.73049128969223721739894670993, −7.21003997088792420629745239650, −6.57369316848351084498912328365, −5.69481051430581996559793048551, −5.11829416356223864781665550292, −4.20441900800865553885759463487, −2.63827358659600944073239821246, −2.18471226823718877240551292181, −0.55315623702950273966156651854, 0,
0.55315623702950273966156651854, 2.18471226823718877240551292181, 2.63827358659600944073239821246, 4.20441900800865553885759463487, 5.11829416356223864781665550292, 5.69481051430581996559793048551, 6.57369316848351084498912328365, 7.21003997088792420629745239650, 7.73049128969223721739894670993
