L(s) = 1 | − 1.43·3-s − 2.23i·5-s + 10.1i·7-s − 6.93·9-s + 13.0i·11-s + 23.2·13-s + 3.21i·15-s + 28.2i·17-s − 11.6i·19-s − 14.6i·21-s + (17.3 − 15.0i)23-s − 5.00·25-s + 22.9·27-s + 42.4·29-s − 18.7·31-s + ⋯ |
L(s) = 1 | − 0.479·3-s − 0.447i·5-s + 1.45i·7-s − 0.770·9-s + 1.18i·11-s + 1.78·13-s + 0.214i·15-s + 1.66i·17-s − 0.614i·19-s − 0.697i·21-s + (0.756 − 0.653i)23-s − 0.200·25-s + 0.848·27-s + 1.46·29-s − 0.605·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 - 0.756i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.653 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.253467646\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.253467646\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-17.3 + 15.0i)T \) |
good | 3 | \( 1 + 1.43T + 9T^{2} \) |
| 7 | \( 1 - 10.1iT - 49T^{2} \) |
| 11 | \( 1 - 13.0iT - 121T^{2} \) |
| 13 | \( 1 - 23.2T + 169T^{2} \) |
| 17 | \( 1 - 28.2iT - 289T^{2} \) |
| 19 | \( 1 + 11.6iT - 361T^{2} \) |
| 29 | \( 1 - 42.4T + 841T^{2} \) |
| 31 | \( 1 + 18.7T + 961T^{2} \) |
| 37 | \( 1 - 1.14iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 72.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 4.96iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 0.813T + 2.20e3T^{2} \) |
| 53 | \( 1 + 26.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 94.0T + 3.48e3T^{2} \) |
| 61 | \( 1 - 74.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 80.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 83.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 8.98T + 5.32e3T^{2} \) |
| 79 | \( 1 + 80.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 94.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 136. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 2.32iT - 9.40e3T^{2} \) |
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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
−8.939845462450136559299659604059, −8.762027592313593537612801172028, −8.088755217478888368589523747265, −6.62387226115738578477407734024, −6.16415989630895858231106697135, −5.36935691492527144878469505774, −4.63926596683199942693909227132, −3.46292294260316304869445561339, −2.34716234857234822707422978951, −1.29767451796897737776188232003,
0.39863439576490086645484120514, 1.23998092803239903190513761429, 3.15893043701812250814009321485, 3.47396614341940882302355250883, 4.72612784858877887932490937922, 5.66273055512391252024018651839, 6.42260714381964993354851493850, 7.03950406245713106599876183527, 8.045762395914526323713083752057, 8.657178886996904052492595339759