L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.5 + 2.59i)7-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)11-s + 3·13-s + (4 − 6.92i)17-s + (0.5 + 0.866i)19-s + (−2 − 1.73i)21-s + (4 + 6.92i)23-s + 0.999·27-s + 4·29-s + (−1.5 + 2.59i)31-s + (−0.999 − 1.73i)33-s + (−0.5 − 0.866i)37-s + (−1.5 + 2.59i)39-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (−0.188 + 0.981i)7-s + (−0.166 − 0.288i)9-s + (−0.301 + 0.522i)11-s + 0.832·13-s + (0.970 − 1.68i)17-s + (0.114 + 0.198i)19-s + (−0.436 − 0.377i)21-s + (0.834 + 1.44i)23-s + 0.192·27-s + 0.742·29-s + (−0.269 + 0.466i)31-s + (−0.174 − 0.301i)33-s + (−0.0821 − 0.142i)37-s + (−0.240 + 0.416i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.425872785\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.425872785\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.5 - 2.59i)T \) |
good | 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 + (-4 + 6.92i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 11T + 43T^{2} \) |
| 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 | \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.5 + 11.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 2T + 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 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
−9.404867300453660491687727090464, −8.756491253430226388308794240206, −7.75881040428835068151252853549, −7.01362081111772859095173615222, −5.96596966402527894882673227731, −5.35265945924216109852996909503, −4.68342841456055938379128876223, −3.39885405141085957937069191497, −2.76403516598285403654098936602, −1.26499302594765782524353377093,
0.59146379977460636388211260175, 1.61882765419548335065548026140, 3.07471891287172468221521771786, 3.86458612992151212977773671285, 4.86976500263577129667346499165, 5.91976027743212472565729749444, 6.48508810262852846364315702894, 7.24994931249509686195318535508, 8.274031303280547287898977168784, 8.481586197286060665889451892708