| L(s) = 1 | − 2.67·3-s + 2.23i·5-s − 10.9i·7-s − 1.84·9-s + 3.70i·11-s − 6.99·13-s − 5.98i·15-s − 14.8i·17-s − 20.1i·19-s + 29.4i·21-s + (13.7 − 18.4i)23-s − 5.00·25-s + 29.0·27-s + 10.7·29-s − 11.9·31-s + ⋯ |
| L(s) = 1 | − 0.891·3-s + 0.447i·5-s − 1.57i·7-s − 0.204·9-s + 0.336i·11-s − 0.537·13-s − 0.398i·15-s − 0.873i·17-s − 1.06i·19-s + 1.40i·21-s + (0.596 − 0.802i)23-s − 0.200·25-s + 1.07·27-s + 0.371·29-s − 0.385·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1396270804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1396270804\) |
| \(L(2)\) |
|
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 | \( 1 + (-13.7 + 18.4i)T \) |
| good | 3 | \( 1 + 2.67T + 9T^{2} \) |
| 7 | \( 1 + 10.9iT - 49T^{2} \) |
| 11 | \( 1 - 3.70iT - 121T^{2} \) |
| 13 | \( 1 + 6.99T + 169T^{2} \) |
| 17 | \( 1 + 14.8iT - 289T^{2} \) |
| 19 | \( 1 + 20.1iT - 361T^{2} \) |
| 29 | \( 1 - 10.7T + 841T^{2} \) |
| 31 | \( 1 + 11.9T + 961T^{2} \) |
| 37 | \( 1 - 32.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 68.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 15.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 3.35T + 2.20e3T^{2} \) |
| 53 | \( 1 + 53.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 95.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 2.07iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 20.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 63.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 95.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 111. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 85.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 73.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 159. iT - 9.40e3T^{2} \) |
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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
−8.553530707179625359143411128635, −7.54016658551609737755988952674, −6.83131106422630001728189629402, −6.52458225156182657482335132076, −5.06248786264423760048156864925, −4.76656801189511687400212309408, −3.55769479187049051180230100488, −2.54850879541766045970189962382, −0.932570029533259381503030257953, −0.05073318577816088937098648914,
1.50387956915056392233023684881, 2.61261629166520642198250716680, 3.74382801147889598734531584987, 5.03596771077478205419637743659, 5.57563888495072809437920626116, 6.01351684151199699912877895092, 7.02720263603944162600566478913, 8.255994951066559369633931462333, 8.635444909550413978213829647449, 9.495035009570266149430958275789
