| L(s) = 1 | + (−1.41 − 1.73i)5-s + (2.44 + i)7-s − 3·9-s + 3.46·11-s + 3.46i·13-s − 4·17-s + 2.82i·19-s − 4.89·23-s + (−0.999 + 4.89i)25-s − 6.92i·29-s − 9.79·31-s + (−1.73 − 5.65i)35-s + 5.65·37-s + 9.79i·41-s − 2.82i·43-s + ⋯ |
| L(s) = 1 | + (−0.632 − 0.774i)5-s + (0.925 + 0.377i)7-s − 9-s + 1.04·11-s + 0.960i·13-s − 0.970·17-s + 0.648i·19-s − 1.02·23-s + (−0.199 + 0.979i)25-s − 1.28i·29-s − 1.75·31-s + (−0.292 − 0.956i)35-s + 0.929·37-s + 1.53i·41-s − 0.431i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6681671269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6681671269\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.41 + 1.73i)T \) |
| 7 | \( 1 + (-2.44 - i)T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 + 4.89T + 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 + 9.79T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 - 9.79iT - 41T^{2} \) |
| 43 | \( 1 + 2.82iT - 43T^{2} \) |
| 47 | \( 1 + 10iT - 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 - 8.48iT - 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 - 14.1iT - 67T^{2} \) |
| 71 | \( 1 - 4iT - 71T^{2} \) |
| 73 | \( 1 + 12T + 73T^{2} \) |
| 79 | \( 1 - 12iT - 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 9.79iT - 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−9.085471287024366854414845348043, −8.513259599227306455790699558894, −8.020471823898631226032302519806, −7.02581713405805312466461113641, −6.04465445103936137350598993116, −5.36516033275033465525790545749, −4.26176454392415867512542618905, −3.94776133353566395257816216595, −2.36406552945223601435610157316, −1.42873940100493837803346371059,
0.22729809404443584290115306947, 1.84339255998328688236901968066, 2.99338588696692329861982467335, 3.81085790908845802243471281131, 4.66694516081669956630298222826, 5.66432238680735487544925876331, 6.50451644777648744305456034391, 7.31046627273451210936245429247, 7.963452219904296854348145873271, 8.689867193825435017330025711114
