| L(s) = 1 | − 1.48i·3-s − 22.0·5-s − 25.0·7-s + 78.7·9-s + 90.5·11-s + 89.9i·13-s + 32.7i·15-s − 493.·17-s + (74.3 + 353. i)19-s + 37.2i·21-s + 781.·23-s − 139.·25-s − 237. i·27-s − 1.27e3i·29-s − 1.62e3i·31-s + ⋯ |
| L(s) = 1 | − 0.164i·3-s − 0.881·5-s − 0.512·7-s + 0.972·9-s + 0.748·11-s + 0.532i·13-s + 0.145i·15-s − 1.70·17-s + (0.205 + 0.978i)19-s + 0.0844i·21-s + 1.47·23-s − 0.222·25-s − 0.325i·27-s − 1.51i·29-s − 1.68i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.274793884\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.274793884\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-74.3 - 353. i)T \) |
| good | 3 | \( 1 + 1.48iT - 81T^{2} \) |
| 5 | \( 1 + 22.0T + 625T^{2} \) |
| 7 | \( 1 + 25.0T + 2.40e3T^{2} \) |
| 11 | \( 1 - 90.5T + 1.46e4T^{2} \) |
| 13 | \( 1 - 89.9iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 493.T + 8.35e4T^{2} \) |
| 23 | \( 1 - 781.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 1.27e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.62e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 2.41e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.67e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 633.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 3.08e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 2.10e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 5.33e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.96e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 1.71e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 3.68e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 2.41e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.01e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 8.00e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 1.18e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 4.00e3iT - 8.85e7T^{2} \) |
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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
−11.12303569051161536268565472297, −9.793347782159139471705147143917, −9.087141220906539657755216316355, −7.85787226638128341292751322481, −7.01172236871345569646152742050, −6.15589909567369906741065235280, −4.40010006641242470882503504523, −3.82817116267790683029296544688, −2.08345162714935445036040818854, −0.46621196256573815760597180583,
1.10126203708777905154778612661, 2.98607039175289041447263202188, 4.10232455705563215381480939523, 5.02126278607684413353764247987, 6.81584145710938191367090977617, 7.08563019263938307454687309814, 8.632081871307523263247077239693, 9.244361579026036049381650123694, 10.50802343475492495874379514758, 11.17816791057359329352372371221
