| L(s) = 1 | − 1.73i·3-s − 2.23·5-s − 9.77·7-s − 2.99·9-s + 16.4i·11-s + 13.2i·13-s + 3.87i·15-s − 30.1i·17-s + 20.5·19-s + 16.9i·21-s + 32.5i·23-s + 5.00·25-s + 5.19i·27-s − 4.89i·29-s + (−29.6 − 9.04i)31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.447·5-s − 1.39·7-s − 0.333·9-s + 1.49i·11-s + 1.01i·13-s + 0.258i·15-s − 1.77i·17-s + 1.08·19-s + 0.806i·21-s + 1.41i·23-s + 0.200·25-s + 0.192i·27-s − 0.168i·29-s + (−0.956 − 0.291i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.291 + 0.956i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6926559698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6926559698\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 + 2.23T \) |
| 31 | \( 1 + (29.6 + 9.04i)T \) |
| good | 7 | \( 1 + 9.77T + 49T^{2} \) |
| 11 | \( 1 - 16.4iT - 121T^{2} \) |
| 13 | \( 1 - 13.2iT - 169T^{2} \) |
| 17 | \( 1 + 30.1iT - 289T^{2} \) |
| 19 | \( 1 - 20.5T + 361T^{2} \) |
| 23 | \( 1 - 32.5iT - 529T^{2} \) |
| 29 | \( 1 + 4.89iT - 841T^{2} \) |
| 37 | \( 1 - 38.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 28.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 60.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 57.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 4.53iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 85.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 39.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 14.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + 48.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + 46.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 39.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 73.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 170. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 163.T + 9.40e3T^{2} \) |
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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
−9.088910343454988599659882121019, −7.72603141019428512209851126196, −7.06383515946167330742026763747, −6.86377951558870727364057950094, −5.62857523482847886718554317320, −4.75162253788209288618744921142, −3.66450368760444899614199410698, −2.82790228588012131592178430947, −1.69477521067564591974411909932, −0.23890457324559225823585255109,
0.835862642696515142324302898594, 2.81508576448213223260937932164, 3.42158978462552060144810984517, 4.06826628346528254301172659927, 5.45623422304120389672941337379, 5.98627731299828391251962858108, 6.76662480168478986904837348293, 7.957910877451185555411241229828, 8.495645846775590757065706865664, 9.291018709230149575780405531558
