| L(s) = 1 | − 2.82i·2-s − 11.2i·3-s − 8.00·4-s + (−5.13 − 24.4i)5-s − 31.8·6-s − 9.84·7-s + 22.6i·8-s − 46.0·9-s + (−69.2 + 14.5i)10-s − 151. i·11-s + 90.1i·12-s − 99.9i·13-s + 27.8i·14-s + (−275. + 57.9i)15-s + 64.0·16-s − 484.·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.25i·3-s − 0.500·4-s + (−0.205 − 0.978i)5-s − 0.885·6-s − 0.200·7-s + 0.353i·8-s − 0.568·9-s + (−0.692 + 0.145i)10-s − 1.25i·11-s + 0.626i·12-s − 0.591i·13-s + 0.142i·14-s + (−1.22 + 0.257i)15-s + 0.250·16-s − 1.67·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0410 - 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0410 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.9771034105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9771034105\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82iT \) |
| 5 | \( 1 + (5.13 + 24.4i)T \) |
| 23 | \( 1 + (129. - 512. i)T \) |
| good | 3 | \( 1 + 11.2iT - 81T^{2} \) |
| 7 | \( 1 + 9.84T + 2.40e3T^{2} \) |
| 11 | \( 1 + 151. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 99.9iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 484.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 61.3iT - 1.30e5T^{2} \) |
| 29 | \( 1 - 1.64e3T + 7.07e5T^{2} \) |
| 31 | \( 1 - 662.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 950.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 235.T + 2.82e6T^{2} \) |
| 43 | \( 1 - 780.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 1.43e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.55e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 5.08e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 435. iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 3.96e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 2.54e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 3.90e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 7.87e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 9.12e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 7.93e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.50e4T + 8.85e7T^{2} \) |
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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
−11.15721317575779889643866032279, −9.847742062274713402853768864484, −8.572811188935509273691040419564, −8.156747352704602626134085002016, −6.72595653748857122369473482923, −5.61208269796733358278105277759, −4.22414717368489002088992341288, −2.69379278387590277556197715719, −1.27548242359124902413796364316, −0.35621491519820681296942233492,
2.59087129665885491806059955165, 4.20951209439209149177007434480, 4.65427313285072217383145881989, 6.41078794228192973903468730835, 7.01709289716175321057104972367, 8.421058630309389844444011306784, 9.499020973176159066464541750302, 10.19007480019352446855507280170, 11.00787384933742186115689879114, 12.18656622130906775105084497555
