| L(s) = 1 | + (−0.104 − 0.994i)2-s + (1.38 − 1.54i)3-s + (−0.978 + 0.207i)4-s + (−1.25 + 1.84i)5-s + (−1.67 − 1.21i)6-s + (−1.06 + 2.42i)7-s + (0.309 + 0.951i)8-s + (−0.136 − 1.29i)9-s + (1.97 + 1.05i)10-s + (3.00 + 1.40i)11-s + (−1.03 + 1.79i)12-s + (−0.112 − 0.155i)13-s + (2.52 + 0.802i)14-s + (1.10 + 4.50i)15-s + (0.913 − 0.406i)16-s + (−2.35 − 0.247i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 − 0.703i)2-s + (0.801 − 0.890i)3-s + (−0.489 + 0.103i)4-s + (−0.562 + 0.826i)5-s + (−0.685 − 0.497i)6-s + (−0.401 + 0.916i)7-s + (0.109 + 0.336i)8-s + (−0.0454 − 0.432i)9-s + (0.623 + 0.334i)10-s + (0.906 + 0.422i)11-s + (−0.299 + 0.518i)12-s + (−0.0312 − 0.0430i)13-s + (0.673 + 0.214i)14-s + (0.285 + 1.16i)15-s + (0.228 − 0.101i)16-s + (−0.570 − 0.0599i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.58757 - 0.113224i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58757 - 0.113224i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 + 0.994i)T \) |
| 5 | \( 1 + (1.25 - 1.84i)T \) |
| 7 | \( 1 + (1.06 - 2.42i)T \) |
| 11 | \( 1 + (-3.00 - 1.40i)T \) |
| good | 3 | \( 1 + (-1.38 + 1.54i)T + (-0.313 - 2.98i)T^{2} \) |
| 13 | \( 1 + (0.112 + 0.155i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (2.35 + 0.247i)T + (16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-5.10 - 1.08i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-2.09 - 1.20i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.20 - 1.69i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.88 - 4.23i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (1.26 - 1.13i)T + (3.86 - 36.7i)T^{2} \) |
| 41 | \( 1 + (-1.81 - 5.58i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + (0.432 + 0.0919i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (0.901 - 2.02i)T + (-35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (1.02 + 4.84i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (12.3 - 5.49i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (7.38 - 4.26i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.70 + 4.87i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (2.20 + 10.3i)T + (-66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (9.70 - 1.02i)T + (77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (0.100 - 0.137i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-3.36 - 1.94i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.277 - 0.201i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.32094708981341812154953772900, −9.280381791719990263959516347236, −8.746158677402002788361992440305, −7.70806372173907783323712074171, −7.05258803939424108934518429220, −6.09754754626162142424367474267, −4.61898836369272820387238164137, −3.26899319911867496559719418018, −2.72893537790693509685737058858, −1.52150624794405519918315933045,
0.842650406308167208436468725715, 3.20295632177527822004481440599, 4.09941883912050280121801453537, 4.59266296678354210221426078012, 5.92985891846717549812593651546, 7.05524490114679603829134844956, 7.82033381503029834088097852163, 8.888394052351458647008559569739, 9.155637778502642249113555728818, 10.00176891049064553610297335015
