| L(s) = 1 | + 0.929·2-s + 3-s − 1.13·4-s + 3.14·5-s + 0.929·6-s − 4.69·7-s − 2.91·8-s + 9-s + 2.92·10-s + 3.66·11-s − 1.13·12-s + 13-s − 4.36·14-s + 3.14·15-s − 0.433·16-s − 1.38·17-s + 0.929·18-s − 7.63·19-s − 3.57·20-s − 4.69·21-s + 3.40·22-s − 6.10·23-s − 2.91·24-s + 4.91·25-s + 0.929·26-s + 27-s + 5.33·28-s + ⋯ |
| L(s) = 1 | + 0.656·2-s + 0.577·3-s − 0.568·4-s + 1.40·5-s + 0.379·6-s − 1.77·7-s − 1.03·8-s + 0.333·9-s + 0.925·10-s + 1.10·11-s − 0.328·12-s + 0.277·13-s − 1.16·14-s + 0.812·15-s − 0.108·16-s − 0.335·17-s + 0.218·18-s − 1.75·19-s − 0.800·20-s − 1.02·21-s + 0.726·22-s − 1.27·23-s − 0.594·24-s + 0.982·25-s + 0.182·26-s + 0.192·27-s + 1.00·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 0.929T + 2T^{2} \) |
| 5 | \( 1 - 3.14T + 5T^{2} \) |
| 7 | \( 1 + 4.69T + 7T^{2} \) |
| 11 | \( 1 - 3.66T + 11T^{2} \) |
| 17 | \( 1 + 1.38T + 17T^{2} \) |
| 19 | \( 1 + 7.63T + 19T^{2} \) |
| 23 | \( 1 + 6.10T + 23T^{2} \) |
| 29 | \( 1 - 7.71T + 29T^{2} \) |
| 31 | \( 1 + 8.72T + 31T^{2} \) |
| 37 | \( 1 + 6.63T + 37T^{2} \) |
| 41 | \( 1 + 5.87T + 41T^{2} \) |
| 43 | \( 1 + 5.93T + 43T^{2} \) |
| 47 | \( 1 + 4.57T + 47T^{2} \) |
| 53 | \( 1 - 3.98T + 53T^{2} \) |
| 59 | \( 1 + 3.82T + 59T^{2} \) |
| 61 | \( 1 - 5.46T + 61T^{2} \) |
| 67 | \( 1 - 2.19T + 67T^{2} \) |
| 71 | \( 1 + 1.33T + 71T^{2} \) |
| 73 | \( 1 + 0.304T + 73T^{2} \) |
| 79 | \( 1 - 7.29T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 - 0.821T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−8.583142732214355969195798524845, −6.91820471093713195101017260213, −6.33660122113497051034416452140, −6.10608120131953679346560424891, −5.08935817066572150404482216458, −4.00450549605002764052502256694, −3.58729189923423453414050083821, −2.62573526004537422219664050015, −1.72475367011571161766735422454, 0,
1.72475367011571161766735422454, 2.62573526004537422219664050015, 3.58729189923423453414050083821, 4.00450549605002764052502256694, 5.08935817066572150404482216458, 6.10608120131953679346560424891, 6.33660122113497051034416452140, 6.91820471093713195101017260213, 8.583142732214355969195798524845
