| 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
−11.63016709402490148704419020299, −10.92718450276541568455388702181, −9.635290310151336322908168568995, −8.454915862625768212448698570053, −8.243436150637159533033902581779, −7.16711165489601213150746912514, −5.47618787654498456789890195535, −4.24377583420079298700634401333, −3.35742191266501768532515309819, −1.58626197392796318237101138798,
0.085241330724985313911238958988, 3.30923207317567307696740101740, 4.24502924671795758150422312853, 5.28408339188929071586896088007, 6.84024504726764710784004106119, 7.26682277579367908440355936365, 8.283541569738879558156426758650, 9.117958196963978038482571939204, 10.45800564526952502988141630034, 10.81106447954353558930448826898
