L(s) = 1 | + 3.09·5-s + i·7-s − 0.646i·11-s + 4.37i·13-s + 0.295i·17-s − 5.29·19-s − 3.94·23-s + 4.58·25-s − 5.03·29-s + 9.16i·31-s + 3.09i·35-s + 2.55i·37-s + 10.4i·41-s − 1.63·43-s − 5.65·47-s + ⋯ |
L(s) = 1 | + 1.38·5-s + 0.377i·7-s − 0.194i·11-s + 1.21i·13-s + 0.0715i·17-s − 1.21·19-s − 0.823·23-s + 0.916·25-s − 0.934·29-s + 1.64i·31-s + 0.523i·35-s + 0.419i·37-s + 1.62i·41-s − 0.249·43-s − 0.825·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.346 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.346 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.684283526\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.684283526\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 - 3.09T + 5T^{2} \) |
| 11 | \( 1 + 0.646iT - 11T^{2} \) |
| 13 | \( 1 - 4.37iT - 13T^{2} \) |
| 17 | \( 1 - 0.295iT - 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 + 3.94T + 23T^{2} \) |
| 29 | \( 1 + 5.03T + 29T^{2} \) |
| 31 | \( 1 - 9.16iT - 31T^{2} \) |
| 37 | \( 1 - 2.55iT - 37T^{2} \) |
| 41 | \( 1 - 10.4iT - 41T^{2} \) |
| 43 | \( 1 + 1.63T + 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 - 8.64T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 2.55T + 67T^{2} \) |
| 71 | \( 1 - 1.11T + 71T^{2} \) |
| 73 | \( 1 + 9.58T + 73T^{2} \) |
| 79 | \( 1 + 16.7iT - 79T^{2} \) |
| 83 | \( 1 - 8.50iT - 83T^{2} \) |
| 89 | \( 1 + 10.4iT - 89T^{2} \) |
| 97 | \( 1 + 2.41T + 97T^{2} \) |
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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
−8.885904144765320950566044441454, −8.076576556686315997620031743124, −6.97183919071908257886166189029, −6.35181172011264581757340100762, −5.86939712537080747169340275090, −4.98164525893579253620329209920, −4.21870929142143143680515880081, −3.08128361200671333776067182104, −2.07331282854320011183176754703, −1.56162107453976556137600948323,
0.42525949503236434345250898525, 1.85634811614904717288564751826, 2.40272719999610866068682169970, 3.59635633457202275810345304346, 4.42799880590052964376184156937, 5.55630877314599363886260082917, 5.80268797472131762186290261592, 6.66078152124407248548061397946, 7.49632542333342411442210707713, 8.223239585722539366913872440886