| L(s) = 1 | + (0.0875 + 0.0234i)2-s + (1.83 − 2.37i)3-s + (−3.45 − 1.99i)4-s + (−4.80 + 1.37i)5-s + (0.216 − 0.164i)6-s + (2.18 − 6.65i)7-s + (−0.512 − 0.512i)8-s + (−2.26 − 8.71i)9-s + (−0.453 + 0.00753i)10-s + (−6.21 − 3.58i)11-s + (−11.0 + 4.53i)12-s + (2.42 + 2.42i)13-s + (0.346 − 0.531i)14-s + (−5.56 + 13.9i)15-s + (7.95 + 13.7i)16-s + (17.1 − 4.60i)17-s + ⋯ |
| L(s) = 1 | + (0.0437 + 0.0117i)2-s + (0.611 − 0.790i)3-s + (−0.864 − 0.498i)4-s + (−0.961 + 0.274i)5-s + (0.0360 − 0.0274i)6-s + (0.311 − 0.950i)7-s + (−0.0640 − 0.0640i)8-s + (−0.251 − 0.967i)9-s + (−0.0453 + 0.000753i)10-s + (−0.565 − 0.326i)11-s + (−0.923 + 0.378i)12-s + (0.186 + 0.186i)13-s + (0.0247 − 0.0379i)14-s + (−0.370 + 0.928i)15-s + (0.496 + 0.860i)16-s + (1.01 − 0.270i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.518433 - 0.915255i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.518433 - 0.915255i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.83 + 2.37i)T \) |
| 5 | \( 1 + (4.80 - 1.37i)T \) |
| 7 | \( 1 + (-2.18 + 6.65i)T \) |
| good | 2 | \( 1 + (-0.0875 - 0.0234i)T + (3.46 + 2i)T^{2} \) |
| 11 | \( 1 + (6.21 + 3.58i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-2.42 - 2.42i)T + 169iT^{2} \) |
| 17 | \( 1 + (-17.1 + 4.60i)T + (250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (6.28 + 10.8i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-0.0951 + 0.355i)T + (-458. - 264.5i)T^{2} \) |
| 29 | \( 1 - 51.7T + 841T^{2} \) |
| 31 | \( 1 + (-2.02 - 1.16i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (4.65 - 17.3i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 - 44.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + (45.6 - 45.6i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-14.6 + 54.7i)T + (-1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (63.8 - 17.1i)T + (2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (65.6 + 37.9i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-93.3 + 53.8i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (29.9 - 8.01i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 69.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-26.1 + 7.00i)T + (4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-68.4 + 39.5i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-77.2 + 77.2i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (18.4 - 10.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (45.5 - 45.5i)T - 9.40e3iT^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.39067596619500632895898558265, −12.32280893583571097897604856732, −11.06809945043749169696057090994, −9.929139124326945756266459297649, −8.490207100362169876403172624518, −7.79957058199540157815364668378, −6.55751204556717456580714290546, −4.70195776150557979593581689566, −3.34512825514494693269208762658, −0.78626538279120194377187048919,
3.06593672298435021985754032764, 4.32170569139952166220141134517, 5.33935053622456494381832465430, 7.88455336041635633363506213248, 8.341283407193153301495164383878, 9.365870085248721806170734664921, 10.57034996965073482860826210417, 12.01958440362080759703370119490, 12.70269413533164818313396882886, 14.04327372179715033382387504352
