| L(s) = 1 | + (0.112 − 0.0725i)2-s + (−0.142 + 0.989i)3-s + (−0.823 + 1.80i)4-s + (0.959 − 0.281i)5-s + (0.0557 + 0.122i)6-s + (0.922 + 1.06i)7-s + (0.0760 + 0.528i)8-s + (−0.959 − 0.281i)9-s + (0.0878 − 0.101i)10-s + (1.08 + 0.696i)11-s + (−1.66 − 1.07i)12-s + (−2.21 + 2.55i)13-s + (0.181 + 0.0532i)14-s + (0.142 + 0.989i)15-s + (−2.54 − 2.94i)16-s + (−0.358 − 0.783i)17-s + ⋯ |
| L(s) = 1 | + (0.0797 − 0.0512i)2-s + (−0.0821 + 0.571i)3-s + (−0.411 + 0.901i)4-s + (0.429 − 0.125i)5-s + (0.0227 + 0.0498i)6-s + (0.348 + 0.402i)7-s + (0.0268 + 0.186i)8-s + (−0.319 − 0.0939i)9-s + (0.0277 − 0.0320i)10-s + (0.326 + 0.209i)11-s + (−0.481 − 0.309i)12-s + (−0.613 + 0.708i)13-s + (0.0484 + 0.0142i)14-s + (0.0367 + 0.255i)15-s + (−0.637 − 0.735i)16-s + (−0.0868 − 0.190i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.255 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.752606 + 0.977249i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.752606 + 0.977249i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.142 - 0.989i)T \) |
| 5 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (-1.41 - 4.58i)T \) |
| good | 2 | \( 1 + (-0.112 + 0.0725i)T + (0.830 - 1.81i)T^{2} \) |
| 7 | \( 1 + (-0.922 - 1.06i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-1.08 - 0.696i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (2.21 - 2.55i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.358 + 0.783i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (1.88 - 4.12i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.0589 - 0.129i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.361 + 2.51i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-1.06 - 0.313i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-5.76 + 1.69i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.0651 + 0.453i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + (-4.49 - 5.19i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-1.68 + 1.94i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (1.79 + 12.5i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (1.63 - 1.04i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-9.88 + 6.35i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (5.44 - 11.9i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-5.91 + 6.82i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-8.41 - 2.47i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.593 + 4.13i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (9.41 - 2.76i)T + (81.6 - 52.4i)T^{2} \) |
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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
−11.90736088976399943434358314369, −10.95503887196409519815179377083, −9.672102584360695304683944436303, −9.143724378920847481851675656806, −8.167649937137389145582187535451, −7.11357349813888841745256803764, −5.72310479989320162139364714102, −4.68098017253805542526223988165, −3.74781820149519973423264222329, −2.26332459697102503539117726434,
0.893159584563573196587493342556, 2.47412952166320262407381422503, 4.37116214616498492298294744755, 5.39714337734530148269044551391, 6.35408490260366575567541488220, 7.25952890433883461553857344171, 8.517426297428550186359837537897, 9.392169817570084684205424320469, 10.49501210539557391490008214741, 10.97854371226371326953887371683
