L(s) = 1 | + (−0.707 − 0.707i)5-s + 0.296·7-s + (−4.14 + 4.14i)11-s + (−3.69 − 3.69i)13-s + 5.57i·17-s + (2.26 − 2.26i)19-s − 0.797i·23-s + 1.00i·25-s + (6.66 − 6.66i)29-s + 2.04i·31-s + (−0.209 − 0.209i)35-s + (2.62 − 2.62i)37-s + 9.97·41-s + (−2.77 − 2.77i)43-s + 5.51·47-s + ⋯ |
L(s) = 1 | + (−0.316 − 0.316i)5-s + 0.112·7-s + (−1.24 + 1.24i)11-s + (−1.02 − 1.02i)13-s + 1.35i·17-s + (0.518 − 0.518i)19-s − 0.166i·23-s + 0.200i·25-s + (1.23 − 1.23i)29-s + 0.367i·31-s + (−0.0354 − 0.0354i)35-s + (0.431 − 0.431i)37-s + 1.55·41-s + (−0.423 − 0.423i)43-s + 0.805·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 + 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.313170637\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.313170637\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.296T + 7T^{2} \) |
| 11 | \( 1 + (4.14 - 4.14i)T - 11iT^{2} \) |
| 13 | \( 1 + (3.69 + 3.69i)T + 13iT^{2} \) |
| 17 | \( 1 - 5.57iT - 17T^{2} \) |
| 19 | \( 1 + (-2.26 + 2.26i)T - 19iT^{2} \) |
| 23 | \( 1 + 0.797iT - 23T^{2} \) |
| 29 | \( 1 + (-6.66 + 6.66i)T - 29iT^{2} \) |
| 31 | \( 1 - 2.04iT - 31T^{2} \) |
| 37 | \( 1 + (-2.62 + 2.62i)T - 37iT^{2} \) |
| 41 | \( 1 - 9.97T + 41T^{2} \) |
| 43 | \( 1 + (2.77 + 2.77i)T + 43iT^{2} \) |
| 47 | \( 1 - 5.51T + 47T^{2} \) |
| 53 | \( 1 + (-7.43 - 7.43i)T + 53iT^{2} \) |
| 59 | \( 1 + (-9.43 + 9.43i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.274 - 0.274i)T + 61iT^{2} \) |
| 67 | \( 1 + (-8.79 + 8.79i)T - 67iT^{2} \) |
| 71 | \( 1 - 0.256iT - 71T^{2} \) |
| 73 | \( 1 + 1.18iT - 73T^{2} \) |
| 79 | \( 1 + 10.9iT - 79T^{2} \) |
| 83 | \( 1 + (10.1 + 10.1i)T + 83iT^{2} \) |
| 89 | \( 1 - 2.04T + 89T^{2} \) |
| 97 | \( 1 - 0.781T + 97T^{2} \) |
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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
−8.554229566560677570971205593325, −7.73166025562317579166484210594, −7.55899420859570069647813638337, −6.42506727678207756452894944094, −5.43853459068976558667345635242, −4.84266513844135272019114152715, −4.10091754078622937350458266237, −2.84185038226105806005274735562, −2.13080972898375319402610584004, −0.56830280764726611539437561502,
0.832915822746408396837234876305, 2.49271066503337836687578883990, 2.99172988539732721705860394278, 4.14110626627077817492242818235, 5.06801655002049588925060064311, 5.62918972055008088415500711230, 6.76968482106029316941020908592, 7.31036489954839364443192449643, 8.083633265911438750798728785749, 8.742716974529381942338062746532