| L(s) = 1 | + (−0.0713 + 0.997i)2-s + (0.977 − 0.212i)3-s + (−0.989 − 0.142i)4-s + (0.389 − 2.20i)5-s + (0.142 + 0.989i)6-s + (−2.31 + 1.26i)7-s + (0.212 − 0.977i)8-s + (0.909 − 0.415i)9-s + (2.16 + 0.545i)10-s + (−3.41 − 2.96i)11-s + (−0.997 + 0.0713i)12-s + (2.21 − 4.05i)13-s + (−1.09 − 2.39i)14-s + (−0.0877 − 2.23i)15-s + (0.959 + 0.281i)16-s + (−3.75 − 2.81i)17-s + ⋯ |
| L(s) = 1 | + (−0.0504 + 0.705i)2-s + (0.564 − 0.122i)3-s + (−0.494 − 0.0711i)4-s + (0.174 − 0.984i)5-s + (0.0580 + 0.404i)6-s + (−0.873 + 0.477i)7-s + (0.0751 − 0.345i)8-s + (0.303 − 0.138i)9-s + (0.685 + 0.172i)10-s + (−1.03 − 0.893i)11-s + (−0.287 + 0.0205i)12-s + (0.614 − 1.12i)13-s + (−0.292 − 0.640i)14-s + (−0.0226 − 0.576i)15-s + (0.239 + 0.0704i)16-s + (−0.910 − 0.681i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.373 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.373 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.981340 - 0.662500i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.981340 - 0.662500i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0713 - 0.997i)T \) |
| 3 | \( 1 + (-0.977 + 0.212i)T \) |
| 5 | \( 1 + (-0.389 + 2.20i)T \) |
| 23 | \( 1 + (-4.79 - 0.0981i)T \) |
| good | 7 | \( 1 + (2.31 - 1.26i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (3.41 + 2.96i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.21 + 4.05i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (3.75 + 2.81i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.166 - 1.15i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-1.94 + 0.280i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-1.55 + 1.00i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (10.9 + 4.10i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (-4.03 + 8.84i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (0.0662 + 0.304i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-3.83 - 3.83i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.21 + 7.71i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (0.914 + 3.11i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-7.55 - 11.7i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-11.7 - 0.841i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-7.31 - 8.43i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.512 - 0.684i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (6.01 - 1.76i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-3.47 + 9.32i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (-11.9 - 7.66i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-3.49 - 9.37i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−10.05870286886430535233044327030, −9.063838731894298558901431561557, −8.637243974067608615154991658485, −7.88450218557571034657748859203, −6.78585049529698817187659417694, −5.70547925049204683543715319409, −5.15793969195620805904806035667, −3.70552355561985752388560101985, −2.61241736783292900135230707834, −0.57780135129366337689116406807,
1.93590392103962353552652033723, 2.92400173604441611626072867830, 3.82456110163615276059606974634, 4.86034946403729823070236913123, 6.50195678479811935181062293453, 7.00267703982968199479613758160, 8.165396078503515181524220892566, 9.183368621587681034182255786311, 9.879082134708613656802706984918, 10.60671619779913885851388750489
