| L(s) = 1 | + 0.517i·2-s + 1.73·4-s − 3.31i·5-s + (2.34 − 1.22i)7-s + 1.93i·8-s + 3·9-s + 1.71·10-s − 1.71·13-s + (0.633 + 1.21i)14-s + 2.46·16-s − 6.40·17-s + 1.55i·18-s + 4.69·19-s − 5.74i·20-s − 1.26·23-s + ⋯ |
| L(s) = 1 | + 0.366i·2-s + 0.866·4-s − 1.48i·5-s + (0.886 − 0.462i)7-s + 0.683i·8-s + 9-s + 0.542·10-s − 0.476·13-s + (0.169 + 0.324i)14-s + 0.616·16-s − 1.55·17-s + 0.366i·18-s + 1.07·19-s − 1.28i·20-s − 0.264·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.11117 - 0.584344i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.11117 - 0.584344i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.34 + 1.22i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.517iT - 2T^{2} \) |
| 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 + 3.31iT - 5T^{2} \) |
| 13 | \( 1 + 1.71T + 13T^{2} \) |
| 17 | \( 1 + 6.40T + 17T^{2} \) |
| 19 | \( 1 - 4.69T + 19T^{2} \) |
| 23 | \( 1 + 1.26T + 23T^{2} \) |
| 29 | \( 1 - 6.17iT - 29T^{2} \) |
| 31 | \( 1 + 9.06iT - 31T^{2} \) |
| 37 | \( 1 + 8.46T + 37T^{2} \) |
| 41 | \( 1 - 1.71T + 41T^{2} \) |
| 43 | \( 1 + 5.93iT - 43T^{2} \) |
| 47 | \( 1 - 9.06iT - 47T^{2} \) |
| 53 | \( 1 - 3.92T + 53T^{2} \) |
| 59 | \( 1 - 6.63iT - 59T^{2} \) |
| 61 | \( 1 - 8.12T + 61T^{2} \) |
| 67 | \( 1 - 7.66T + 67T^{2} \) |
| 71 | \( 1 + 7.46T + 71T^{2} \) |
| 73 | \( 1 + 4.69T + 73T^{2} \) |
| 79 | \( 1 - 9.14iT - 79T^{2} \) |
| 83 | \( 1 + 9.38T + 83T^{2} \) |
| 89 | \( 1 - 5.74iT - 89T^{2} \) |
| 97 | \( 1 - 8.17iT - 97T^{2} \) |
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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
−10.11323185526371426231772634427, −9.127381475212075485819632331797, −8.335661813160798042063173162380, −7.47709103756953107769331947466, −6.91845285757839282900342130569, −5.57223195729416551018294333097, −4.84674441248610916358285955097, −4.03814932450991213324232421241, −2.15439062454151339559702459898, −1.18946701198994881191402208424,
1.77203356763031668770602929452, 2.55381067821782974734638119446, 3.61721424788070023072287814198, 4.85443243872610256986894827297, 6.12851745785362253612356608856, 7.06493499417281079453745072632, 7.29616033240245354364330337390, 8.517898924533316201577906432537, 9.824049060550355470396865767739, 10.36162281375745238154908262211
