L(s) = 1 | + (−0.5 + 0.866i)3-s + (1 − 1.73i)4-s + (2.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (−3 + 5.19i)11-s + (0.999 + 1.73i)12-s + 5·13-s + (−1.99 − 3.46i)16-s + (3 − 5.19i)17-s + (0.5 + 0.866i)19-s + (−2 + 1.73i)21-s + (0.5 + 0.866i)23-s + (2.5 − 4.33i)25-s + 0.999·27-s + (4 − 3.46i)28-s + 6·29-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (0.5 − 0.866i)4-s + (0.944 + 0.327i)7-s + (−0.166 − 0.288i)9-s + (−0.904 + 1.56i)11-s + (0.288 + 0.499i)12-s + 1.38·13-s + (−0.499 − 0.866i)16-s + (0.727 − 1.26i)17-s + (0.114 + 0.198i)19-s + (−0.436 + 0.377i)21-s + (0.104 + 0.180i)23-s + (0.5 − 0.866i)25-s + 0.192·27-s + (0.755 − 0.654i)28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.60502 + 0.101855i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.60502 + 0.101855i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−10.87699742890481836393325049772, −10.24188879175280374370204475737, −9.473331339187304138948343335740, −8.310238398719040413340336024257, −7.28709521874879857386142411812, −6.23549363334136228540821825558, −5.14138898611262881020016876981, −4.67814758332579250034171209939, −2.81032073272351323418807827555, −1.41228084486232279061352243294,
1.33050985624522978346679624229, 2.93954750076820013256659025927, 3.98305005149303116711957671665, 5.53684978854764559278433097917, 6.29181313736809311815566810704, 7.51641584421071846187434081491, 8.215957292124443851724440526804, 8.666904667437368768231839564594, 10.59552810844395674561022520822, 11.04874881047402161388100752147