L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.499 + 0.866i)12-s + 0.999·15-s + (−0.499 + 0.866i)16-s + (1 + 1.73i)17-s + (−0.5 + 0.866i)19-s + 0.999·20-s − 27-s − 29-s + 41-s + 0.999·48-s + (0.999 − 1.73i)51-s + (0.5 + 0.866i)53-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.499 + 0.866i)12-s + 0.999·15-s + (−0.499 + 0.866i)16-s + (1 + 1.73i)17-s + (−0.5 + 0.866i)19-s + 0.999·20-s − 27-s − 29-s + 41-s + 0.999·48-s + (0.999 − 1.73i)51-s + (0.5 + 0.866i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6952549881\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6952549881\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 59 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 3 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - 2T + T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.970734531638442034114979246400, −8.024103983984996056450849366681, −7.45238127489098577839609074551, −6.56336117541269695721486973867, −6.00704660709393530079812375090, −5.47614292994195526031916082798, −4.11667080419362854687182880336, −3.57013077139463056624510474976, −2.04438277647124221582061458663, −1.14196562623647906059293785993,
0.56076330969370378901960868609, 2.50869023593002593909654373793, 3.62048329926210282752191082381, 4.26795751869065964106450835222, 5.04318682543637368771412246906, 5.32302814214158257546036957263, 6.78816166163217476550127867116, 7.62739210940015870500670592921, 8.137850191310251402014417204512, 9.135966634529540245822547923456