L(s) = 1 | + (−0.707 − 0.707i)5-s − 4.02i·7-s + (−0.646 − 0.646i)11-s + (4.91 − 4.91i)13-s + 2.70·17-s + (0.438 − 0.438i)19-s − 3.60i·23-s + 1.00i·25-s + (−2.00 + 2.00i)29-s − 4.30·31-s + (−2.84 + 2.84i)35-s + (−0.743 − 0.743i)37-s − 0.603i·41-s + (5.03 + 5.03i)43-s + 10.8·47-s + ⋯ |
L(s) = 1 | + (−0.316 − 0.316i)5-s − 1.52i·7-s + (−0.195 − 0.195i)11-s + (1.36 − 1.36i)13-s + 0.656·17-s + (0.100 − 0.100i)19-s − 0.750i·23-s + 0.200i·25-s + (−0.373 + 0.373i)29-s − 0.774·31-s + (−0.481 + 0.481i)35-s + (−0.122 − 0.122i)37-s − 0.0943i·41-s + (0.767 + 0.767i)43-s + 1.57·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.535918867\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.535918867\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 + 4.02iT - 7T^{2} \) |
| 11 | \( 1 + (0.646 + 0.646i)T + 11iT^{2} \) |
| 13 | \( 1 + (-4.91 + 4.91i)T - 13iT^{2} \) |
| 17 | \( 1 - 2.70T + 17T^{2} \) |
| 19 | \( 1 + (-0.438 + 0.438i)T - 19iT^{2} \) |
| 23 | \( 1 + 3.60iT - 23T^{2} \) |
| 29 | \( 1 + (2.00 - 2.00i)T - 29iT^{2} \) |
| 31 | \( 1 + 4.30T + 31T^{2} \) |
| 37 | \( 1 + (0.743 + 0.743i)T + 37iT^{2} \) |
| 41 | \( 1 + 0.603iT - 41T^{2} \) |
| 43 | \( 1 + (-5.03 - 5.03i)T + 43iT^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + (4.07 + 4.07i)T + 53iT^{2} \) |
| 59 | \( 1 + (-1.22 - 1.22i)T + 59iT^{2} \) |
| 61 | \( 1 + (6.98 - 6.98i)T - 61iT^{2} \) |
| 67 | \( 1 + (5.24 - 5.24i)T - 67iT^{2} \) |
| 71 | \( 1 + 13.7iT - 71T^{2} \) |
| 73 | \( 1 - 1.30iT - 73T^{2} \) |
| 79 | \( 1 + 0.611T + 79T^{2} \) |
| 83 | \( 1 + (-1.29 + 1.29i)T - 83iT^{2} \) |
| 89 | \( 1 + 10.9iT - 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−8.380100164253408989687936198491, −7.70567800105031534434147129927, −7.21811055615128267127689478945, −6.13852387296515953594931062704, −5.46838117231996121577450307760, −4.42290115934927625457144089522, −3.71825955801444165473167277776, −3.02154418956695063551204959732, −1.32388391976543762356404584582, −0.53538844710510925036933647822,
1.52430991053254894532299044337, 2.43423251355292341613341908564, 3.46818932225154102577114527915, 4.22190021342095894519507797605, 5.42510201482606007858162144290, 5.89826971462876759550643730696, 6.71956856540174982122116880148, 7.59734598431589432959338467309, 8.361849760500393826296445208136, 9.159776199828504641466023860206