| L(s) = 1 | + (−0.470 + 1.33i)2-s + (−0.0980 + 0.995i)3-s + (−1.55 − 1.25i)4-s + (−3.58 + 1.91i)5-s + (−1.28 − 0.599i)6-s + (−0.222 − 1.12i)7-s + (2.40 − 1.48i)8-s + (−0.980 − 0.195i)9-s + (−0.868 − 5.67i)10-s + (−0.503 + 0.613i)11-s + (1.40 − 1.42i)12-s + (0.698 − 1.30i)13-s + (1.59 + 0.229i)14-s + (−1.55 − 3.75i)15-s + (0.849 + 3.90i)16-s + (0.594 − 1.43i)17-s + ⋯ |
| L(s) = 1 | + (−0.332 + 0.943i)2-s + (−0.0565 + 0.574i)3-s + (−0.778 − 0.627i)4-s + (−1.60 + 0.856i)5-s + (−0.523 − 0.244i)6-s + (−0.0842 − 0.423i)7-s + (0.850 − 0.525i)8-s + (−0.326 − 0.0650i)9-s + (−0.274 − 1.79i)10-s + (−0.151 + 0.184i)11-s + (0.404 − 0.411i)12-s + (0.193 − 0.362i)13-s + (0.427 + 0.0614i)14-s + (−0.401 − 0.969i)15-s + (0.212 + 0.977i)16-s + (0.144 − 0.348i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0920343 - 0.0594643i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0920343 - 0.0594643i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.470 - 1.33i)T \) |
| 3 | \( 1 + (0.0980 - 0.995i)T \) |
| good | 5 | \( 1 + (3.58 - 1.91i)T + (2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (0.222 + 1.12i)T + (-6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (0.503 - 0.613i)T + (-2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.698 + 1.30i)T + (-7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-0.594 + 1.43i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (1.53 + 5.07i)T + (-15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (-0.918 + 0.613i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (3.87 - 3.17i)T + (5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (2.12 - 2.12i)T - 31iT^{2} \) |
| 37 | \( 1 + (10.0 + 3.06i)T + (30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (5.54 + 8.29i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.767 - 7.78i)T + (-42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (3.90 + 1.61i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (5.45 + 4.48i)T + (10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (-6.80 - 12.7i)T + (-32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (1.98 + 0.195i)T + (59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (5.65 + 0.557i)T + (65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (-2.15 + 0.427i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-2.48 + 12.4i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (10.1 - 4.22i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-0.157 + 0.0478i)T + (69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (13.6 + 9.10i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (5.98 - 5.98i)T - 97iT^{2} \) |
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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
−10.81106447954353558930448826898, −10.45800564526952502988141630034, −9.117958196963978038482571939204, −8.283541569738879558156426758650, −7.26682277579367908440355936365, −6.84024504726764710784004106119, −5.28408339188929071586896088007, −4.24502924671795758150422312853, −3.30923207317567307696740101740, −0.085241330724985313911238958988,
1.58626197392796318237101138798, 3.35742191266501768532515309819, 4.24377583420079298700634401333, 5.47618787654498456789890195535, 7.16711165489601213150746912514, 8.243436150637159533033902581779, 8.454915862625768212448698570053, 9.635290310151336322908168568995, 10.92718450276541568455388702181, 11.63016709402490148704419020299
