| 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 | \( 1 + (45.5 + 45.5i)T + 9.40e3iT^{2} \) |
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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
−14.04327372179715033382387504352, −12.70269413533164818313396882886, −12.01958440362080759703370119490, −10.57034996965073482860826210417, −9.365870085248721806170734664921, −8.341283407193153301495164383878, −7.88455336041635633363506213248, −5.33935053622456494381832465430, −4.32170569139952166220141134517, −3.06593672298435021985754032764,
0.78626538279120194377187048919, 3.34512825514494693269208762658, 4.70195776150557979593581689566, 6.55751204556717456580714290546, 7.79957058199540157815364668378, 8.490207100362169876403172624518, 9.929139124326945756266459297649, 11.06809945043749169696057090994, 12.32280893583571097897604856732, 13.39067596619500632895898558265
