L(s) = 1 | + 2.64·3-s − 1.73i·5-s + 4.00·9-s − 4.58i·11-s + 3.46i·13-s − 4.58i·15-s − 5.19i·17-s + 2.64·19-s + 4.58i·23-s + 2.00·25-s + 2.64·27-s − 2.64·31-s − 12.1i·33-s + 7·37-s + 9.16i·39-s + ⋯ |
L(s) = 1 | + 1.52·3-s − 0.774i·5-s + 1.33·9-s − 1.38i·11-s + 0.960i·13-s − 1.18i·15-s − 1.26i·17-s + 0.606·19-s + 0.955i·23-s + 0.400·25-s + 0.509·27-s − 0.475·31-s − 2.11i·33-s + 1.15·37-s + 1.46i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.39893 - 0.894383i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.39893 - 0.894383i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2.64T + 3T^{2} \) |
| 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 11 | \( 1 + 4.58iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 5.19iT - 17T^{2} \) |
| 19 | \( 1 - 2.64T + 19T^{2} \) |
| 23 | \( 1 - 4.58iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 2.64T + 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 9.16iT - 43T^{2} \) |
| 47 | \( 1 + 7.93T + 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + 7.93T + 59T^{2} \) |
| 61 | \( 1 - 1.73iT - 61T^{2} \) |
| 67 | \( 1 - 4.58iT - 67T^{2} \) |
| 71 | \( 1 - 9.16iT - 71T^{2} \) |
| 73 | \( 1 - 5.19iT - 73T^{2} \) |
| 79 | \( 1 - 4.58iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 1.73iT - 89T^{2} \) |
| 97 | \( 1 + 3.46iT - 97T^{2} \) |
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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
−9.663860725406179085747627895606, −9.278305293491290063983493604755, −8.587448687362488269378522988300, −7.87274332310234404532035344899, −6.99338158235620623193801213627, −5.68195131611351727254162070929, −4.59423833393748480326918488903, −3.51614842757731313082074368696, −2.68400497140023709354755558156, −1.24265905466638619844136514880,
1.87264022378996563035479341153, 2.82784359114572337154693213946, 3.66027575408347676145119969676, 4.75585728871569662092256299835, 6.20678833967061048478611712486, 7.22650335556974889495287054308, 7.83821062609616349995624810913, 8.610558489760635204837870951647, 9.563882351279743273455760844042, 10.21666436426837769871106144216