| L(s) = 1 | + (−1.66 + 2.28i)2-s + (−1.86 + 5.75i)3-s + (−2.47 − 7.60i)4-s + (−23.9 + 17.3i)5-s + (−10.0 − 13.8i)6-s + (−38.7 + 12.5i)7-s + (21.5 + 6.99i)8-s + (35.9 + 26.0i)9-s − 83.7i·10-s + (107. + 54.9i)11-s + 48.4·12-s + (40.3 − 55.4i)13-s + (35.5 − 109. i)14-s + (−55.3 − 170. i)15-s + (−51.7 + 37.6i)16-s + (207. + 286. i)17-s + ⋯ |
| L(s) = 1 | + (−0.415 + 0.572i)2-s + (−0.207 + 0.639i)3-s + (−0.154 − 0.475i)4-s + (−0.957 + 0.695i)5-s + (−0.279 − 0.384i)6-s + (−0.790 + 0.256i)7-s + (0.336 + 0.109i)8-s + (0.443 + 0.322i)9-s − 0.837i·10-s + (0.891 + 0.453i)11-s + 0.336·12-s + (0.238 − 0.328i)13-s + (0.181 − 0.558i)14-s + (−0.245 − 0.756i)15-s + (−0.202 + 0.146i)16-s + (0.719 + 0.990i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.514i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.189367 + 0.683483i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.189367 + 0.683483i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.66 - 2.28i)T \) |
| 11 | \( 1 + (-107. - 54.9i)T \) |
| good | 3 | \( 1 + (1.86 - 5.75i)T + (-65.5 - 47.6i)T^{2} \) |
| 5 | \( 1 + (23.9 - 17.3i)T + (193. - 594. i)T^{2} \) |
| 7 | \( 1 + (38.7 - 12.5i)T + (1.94e3 - 1.41e3i)T^{2} \) |
| 13 | \( 1 + (-40.3 + 55.4i)T + (-8.82e3 - 2.71e4i)T^{2} \) |
| 17 | \( 1 + (-207. - 286. i)T + (-2.58e4 + 7.94e4i)T^{2} \) |
| 19 | \( 1 + (171. + 55.8i)T + (1.05e5 + 7.66e4i)T^{2} \) |
| 23 | \( 1 + 686.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (-1.23e3 + 401. i)T + (5.72e5 - 4.15e5i)T^{2} \) |
| 31 | \( 1 + (-257. - 186. i)T + (2.85e5 + 8.78e5i)T^{2} \) |
| 37 | \( 1 + (644. + 1.98e3i)T + (-1.51e6 + 1.10e6i)T^{2} \) |
| 41 | \( 1 + (-2.40e3 - 779. i)T + (2.28e6 + 1.66e6i)T^{2} \) |
| 43 | \( 1 - 749. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (955. - 2.94e3i)T + (-3.94e6 - 2.86e6i)T^{2} \) |
| 53 | \( 1 + (-2.67e3 - 1.94e3i)T + (2.43e6 + 7.50e6i)T^{2} \) |
| 59 | \( 1 + (1.90e3 + 5.85e3i)T + (-9.80e6 + 7.12e6i)T^{2} \) |
| 61 | \( 1 + (-1.37e3 - 1.89e3i)T + (-4.27e6 + 1.31e7i)T^{2} \) |
| 67 | \( 1 - 6.27e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (-382. + 277. i)T + (7.85e6 - 2.41e7i)T^{2} \) |
| 73 | \( 1 + (-3.78e3 + 1.23e3i)T + (2.29e7 - 1.66e7i)T^{2} \) |
| 79 | \( 1 + (-5.58e3 + 7.69e3i)T + (-1.20e7 - 3.70e7i)T^{2} \) |
| 83 | \( 1 + (-422. - 580. i)T + (-1.46e7 + 4.51e7i)T^{2} \) |
| 89 | \( 1 + 5.92e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-1.15e3 - 840. i)T + (2.73e7 + 8.41e7i)T^{2} \) |
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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
−17.61130258772248324750928177620, −16.20057692501659326986853371218, −15.55465452480352663056927723218, −14.44614921509850286642752167199, −12.43710150305105589649475825296, −10.80272403635376638653819576650, −9.676396227136495360015039038680, −7.86918244612796617663167422298, −6.33290788353945221745312127617, −4.01349774258012008038434699937,
0.76020164479704840167870810603, 3.90902538062603142194163316340, 6.76057117765068669546422042598, 8.344249138202620347474449482300, 9.844955799203662713061873152662, 11.79143571023969124514179932554, 12.35352435474519663689238306504, 13.77260441829884508893486393131, 15.88832677038661295918463688511, 16.73346758927419222387115007037
