L(s) = 1 | + (0.707 + 0.707i)5-s + 0.511i·7-s + (−0.751 − 0.751i)11-s + (−4.22 + 4.22i)13-s + 4.73·17-s + (4.78 − 4.78i)19-s + 0.0927i·23-s + 1.00i·25-s + (−0.979 + 0.979i)29-s + 3.03·31-s + (−0.361 + 0.361i)35-s + (1.17 + 1.17i)37-s + 11.9i·41-s + (−6.52 − 6.52i)43-s − 5.91·47-s + ⋯ |
L(s) = 1 | + (0.316 + 0.316i)5-s + 0.193i·7-s + (−0.226 − 0.226i)11-s + (−1.17 + 1.17i)13-s + 1.14·17-s + (1.09 − 1.09i)19-s + 0.0193i·23-s + 0.200i·25-s + (−0.181 + 0.181i)29-s + 0.545·31-s + (−0.0611 + 0.0611i)35-s + (0.192 + 0.192i)37-s + 1.86i·41-s + (−0.995 − 0.995i)43-s − 0.863·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.678909328\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.678909328\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.511iT - 7T^{2} \) |
| 11 | \( 1 + (0.751 + 0.751i)T + 11iT^{2} \) |
| 13 | \( 1 + (4.22 - 4.22i)T - 13iT^{2} \) |
| 17 | \( 1 - 4.73T + 17T^{2} \) |
| 19 | \( 1 + (-4.78 + 4.78i)T - 19iT^{2} \) |
| 23 | \( 1 - 0.0927iT - 23T^{2} \) |
| 29 | \( 1 + (0.979 - 0.979i)T - 29iT^{2} \) |
| 31 | \( 1 - 3.03T + 31T^{2} \) |
| 37 | \( 1 + (-1.17 - 1.17i)T + 37iT^{2} \) |
| 41 | \( 1 - 11.9iT - 41T^{2} \) |
| 43 | \( 1 + (6.52 + 6.52i)T + 43iT^{2} \) |
| 47 | \( 1 + 5.91T + 47T^{2} \) |
| 53 | \( 1 + (-6.44 - 6.44i)T + 53iT^{2} \) |
| 59 | \( 1 + (-7.60 - 7.60i)T + 59iT^{2} \) |
| 61 | \( 1 + (2.76 - 2.76i)T - 61iT^{2} \) |
| 67 | \( 1 + (4.94 - 4.94i)T - 67iT^{2} \) |
| 71 | \( 1 - 9.80iT - 71T^{2} \) |
| 73 | \( 1 - 5.10iT - 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 + (8.04 - 8.04i)T - 83iT^{2} \) |
| 89 | \( 1 - 2.41iT - 89T^{2} \) |
| 97 | \( 1 + 4.51T + 97T^{2} \) |
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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
−9.020675997501695005451665151381, −8.167843667244941417754873820561, −7.22131372921018469482213097482, −6.87943232234695485857851939091, −5.76802128726197749079258090108, −5.13534289206495807621563389869, −4.29208719752587586829534487879, −3.09317359372583639820379306647, −2.45478100768911519224656072267, −1.17874520119006069201052931752,
0.58324962174292772756858521994, 1.82497844965786411876145067512, 2.95841906972793646997375676849, 3.72731468579011318781475317958, 5.01150179312268619098970310495, 5.36371614138746503652338774074, 6.21313106947121222213569268659, 7.40307074616473082104429492331, 7.72416975387467695750347733671, 8.510373878259690136127081449759