L(s) = 1 | + (1.20 − 0.742i)2-s + (0.898 − 1.78i)4-s + (2.06 + 0.860i)5-s + 0.707·7-s + (−0.244 − 2.81i)8-s + (3.12 − 0.495i)10-s + (−1.79 + 1.79i)11-s + (3.86 − 3.86i)13-s + (0.851 − 0.524i)14-s + (−2.38 − 3.21i)16-s + 0.244i·17-s + (1.53 + 1.53i)19-s + (3.39 − 2.91i)20-s + (−0.831 + 3.50i)22-s − 6.92·23-s + ⋯ |
L(s) = 1 | + (0.851 − 0.524i)2-s + (0.449 − 0.893i)4-s + (0.922 + 0.384i)5-s + 0.267·7-s + (−0.0863 − 0.996i)8-s + (0.987 − 0.156i)10-s + (−0.542 + 0.542i)11-s + (1.07 − 1.07i)13-s + (0.227 − 0.140i)14-s + (−0.596 − 0.802i)16-s + 0.0593i·17-s + (0.352 + 0.352i)19-s + (0.758 − 0.651i)20-s + (−0.177 + 0.746i)22-s − 1.44·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.61984 - 1.32157i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.61984 - 1.32157i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.20 + 0.742i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.06 - 0.860i)T \) |
good | 7 | \( 1 - 0.707T + 7T^{2} \) |
| 11 | \( 1 + (1.79 - 1.79i)T - 11iT^{2} \) |
| 13 | \( 1 + (-3.86 + 3.86i)T - 13iT^{2} \) |
| 17 | \( 1 - 0.244iT - 17T^{2} \) |
| 19 | \( 1 + (-1.53 - 1.53i)T + 19iT^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 + (-4.89 - 4.89i)T + 29iT^{2} \) |
| 31 | \( 1 - 7.60T + 31T^{2} \) |
| 37 | \( 1 + (8.47 + 8.47i)T + 37iT^{2} \) |
| 41 | \( 1 + 2.12iT - 41T^{2} \) |
| 43 | \( 1 + (0.684 + 0.684i)T + 43iT^{2} \) |
| 47 | \( 1 - 4.47iT - 47T^{2} \) |
| 53 | \( 1 + (1.47 + 1.47i)T + 53iT^{2} \) |
| 59 | \( 1 + (5.86 - 5.86i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.0537 - 0.0537i)T + 61iT^{2} \) |
| 67 | \( 1 + (7.85 - 7.85i)T - 67iT^{2} \) |
| 71 | \( 1 - 2.08iT - 71T^{2} \) |
| 73 | \( 1 - 9.69T + 73T^{2} \) |
| 79 | \( 1 + 7.34T + 79T^{2} \) |
| 83 | \( 1 + (-6.80 + 6.80i)T - 83iT^{2} \) |
| 89 | \( 1 - 3.07iT - 89T^{2} \) |
| 97 | \( 1 + 1.39iT - 97T^{2} \) |
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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
−10.37965175904495970393502218114, −9.892385258373606344540272140230, −8.654195798814874656066320693010, −7.53024182159804132484667746007, −6.39065337403975119671581120283, −5.72345310567139408782269282694, −4.89574763123730921499295226299, −3.60732975802527498965077085475, −2.59970612449217845623728407121, −1.44552277755968915227817620471,
1.76560693945760206571859015000, 3.03419558647937244212089601329, 4.33129761601827885108746739109, 5.13037044264961906780020952538, 6.18595185929578506152795530351, 6.57500437087481456159335021391, 8.080864387600572377906925491231, 8.507477116586971182146418689963, 9.638448231087145430410458933754, 10.62361173494332023030849735978