L(s) = 1 | + (−0.5 − 0.866i)2-s + (1.51 − 0.844i)3-s + (−0.499 + 0.866i)4-s + (0.511 + 0.295i)5-s + (−1.48 − 0.887i)6-s + (1.57 − 2.12i)7-s + 0.999·8-s + (1.57 − 2.55i)9-s − 0.590i·10-s + 6.11·11-s + (−0.0248 + 1.73i)12-s + (−1.86 + 3.08i)13-s + (−2.62 − 0.304i)14-s + (1.02 + 0.0146i)15-s + (−0.5 − 0.866i)16-s + (−3.82 + 6.62i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.873 − 0.487i)3-s + (−0.249 + 0.433i)4-s + (0.228 + 0.131i)5-s + (−0.607 − 0.362i)6-s + (0.596 − 0.802i)7-s + 0.353·8-s + (0.524 − 0.851i)9-s − 0.186i·10-s + 1.84·11-s + (−0.00716 + 0.499i)12-s + (−0.518 + 0.855i)13-s + (−0.702 − 0.0814i)14-s + (0.263 + 0.00378i)15-s + (−0.125 − 0.216i)16-s + (−0.927 + 1.60i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.44609 - 1.11338i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44609 - 1.11338i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-1.51 + 0.844i)T \) |
| 7 | \( 1 + (-1.57 + 2.12i)T \) |
| 13 | \( 1 + (1.86 - 3.08i)T \) |
good | 5 | \( 1 + (-0.511 - 0.295i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 - 6.11T + 11T^{2} \) |
| 17 | \( 1 + (3.82 - 6.62i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + 3.95T + 19T^{2} \) |
| 23 | \( 1 + (-7.26 + 4.19i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.39 - 0.803i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.38 + 2.40i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.93 - 1.11i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.36 - 0.787i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.90 + 5.03i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.94 + 2.85i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.30 - 1.90i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.88 - 2.24i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 - 0.100iT - 61T^{2} \) |
| 67 | \( 1 - 10.3iT - 67T^{2} \) |
| 71 | \( 1 + (0.875 + 1.51i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.41 - 9.37i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.17 - 5.50i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9.07iT - 83T^{2} \) |
| 89 | \( 1 + (3.56 - 2.05i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.82 - 4.89i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.60243107179829782118095425746, −9.651651804100490688566275975389, −8.762763168915309150481687032018, −8.319216638107427853396336713353, −6.87506864568497890474039420520, −6.60891865121393855254517121215, −4.33053854235898182145661931144, −3.90040411612903148168152055094, −2.24500120616014495103242823337, −1.35362472173947099910960236988,
1.70457420766540552689285367648, 3.11315445004864470543981300636, 4.56688561983859281352457434281, 5.27778605843825278018021738993, 6.62487149624221331223354282984, 7.50197044717401293337187855654, 8.534424164454124081747646454994, 9.282427449532903792339198151550, 9.485324059213140333569054484158, 10.91718394098080692458082815718