| L(s) = 1 | + (−0.309 − 0.951i)2-s + (0.669 + 0.743i)3-s + (−0.809 + 0.587i)4-s + (0.5 + 0.866i)5-s + (0.499 − 0.866i)6-s + (0.375 + 3.56i)7-s + (0.809 + 0.587i)8-s + (−0.104 + 0.994i)9-s + (0.669 − 0.743i)10-s + (2.46 + 1.09i)11-s + (−0.978 − 0.207i)12-s + (3.21 − 0.683i)13-s + (3.27 − 1.45i)14-s + (−0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (−6.03 + 2.68i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (0.386 + 0.429i)3-s + (−0.404 + 0.293i)4-s + (0.223 + 0.387i)5-s + (0.204 − 0.353i)6-s + (0.141 + 1.34i)7-s + (0.286 + 0.207i)8-s + (−0.0348 + 0.331i)9-s + (0.211 − 0.235i)10-s + (0.743 + 0.330i)11-s + (−0.282 − 0.0600i)12-s + (0.892 − 0.189i)13-s + (0.876 − 0.390i)14-s + (−0.0797 + 0.245i)15-s + (0.0772 − 0.237i)16-s + (−1.46 + 0.651i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.14587 + 0.872741i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14587 + 0.872741i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 3 | \( 1 + (-0.669 - 0.743i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-3.38 - 4.42i)T \) |
| good | 7 | \( 1 + (-0.375 - 3.56i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (-2.46 - 1.09i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-3.21 + 0.683i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (6.03 - 2.68i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (4.82 + 1.02i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (5.88 + 4.27i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.732 - 2.25i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (4.37 - 7.57i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.52 + 2.80i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (-11.0 - 2.35i)T + (39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (-0.373 + 1.14i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.683 + 6.50i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-6.68 - 7.42i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + 8.69T + 61T^{2} \) |
| 67 | \( 1 + (-2.52 - 4.36i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.818 + 7.78i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (-1.24 - 0.555i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (-11.6 + 5.17i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (6.51 - 7.23i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-10.7 + 7.82i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (6.58 - 4.78i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.39960163249535534539632001115, −9.279342328435572099532769532897, −8.702900695172375526819395789123, −8.275887011525536208165578832395, −6.66117468074004811463012256407, −6.00484101521655905526284931819, −4.67404124206715587942875622998, −3.85382798381771672073158349499, −2.60791092491968500519322887436, −1.88713230525327384796079252441,
0.70077264201867375418182935054, 1.99677182099217957298138691356, 3.92834763748452232728209984444, 4.31988392783979706172450523785, 5.90257547499562223424765508062, 6.53657154941074347458533086720, 7.38081269067545833952309369142, 8.164607923190591958314856875865, 8.948698010930937411247455455519, 9.604982353680262841207901465682
