| L(s) = 1 | + (1 − i)2-s − 2i·4-s + 2i·5-s − 2·7-s + (−2 − 2i)8-s + (2 + 2i)10-s + 4i·13-s + (−2 + 2i)14-s − 4·16-s + 2·17-s − 4i·19-s + 4·20-s − 4·23-s + 25-s + (4 + 4i)26-s + ⋯ |
| L(s) = 1 | + (0.707 − 0.707i)2-s − i·4-s + 0.894i·5-s − 0.755·7-s + (−0.707 − 0.707i)8-s + (0.632 + 0.632i)10-s + 1.10i·13-s + (−0.534 + 0.534i)14-s − 16-s + 0.485·17-s − 0.917i·19-s + 0.894·20-s − 0.834·23-s + 0.200·25-s + (0.784 + 0.784i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.09661 - 0.454231i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09661 - 0.454231i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 + i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−14.27967789830314087067313413679, −13.50751769492664210204676631508, −12.27510916837260584493795738950, −11.29820456022791443126402878567, −10.24089300325458465964867268631, −9.232706728984305294883789634900, −7.04757015115889787904429590464, −5.99689698164512799731262664257, −4.13547310437713837033869561692, −2.65049428307336927197666154875,
3.42160512196739537031611417666, 5.05733945414218989480480092363, 6.19224423072169904665005262741, 7.73033600871715252944656625446, 8.780433665194772995003156464804, 10.22129357345684336042357840687, 12.08350832093766156717979241911, 12.70063404200723784944113138791, 13.63584948772805157440499549654, 14.82928336017101520461775056403
