L(s) = 1 | + (2.22 + 0.224i)5-s + (−1.44 − 1.44i)7-s − 2.44i·11-s + (2 + 2i)13-s + (0.449 − 0.449i)17-s + 2·19-s + (0.449 − 0.449i)23-s + (4.89 + i)25-s + 0.449i·29-s − 8.89i·31-s + (−2.89 − 3.55i)35-s + (2.89 − 2.89i)37-s − 4.89·41-s + (6 − 6i)43-s + (9.34 + 9.34i)47-s + ⋯ |
L(s) = 1 | + (0.994 + 0.100i)5-s + (−0.547 − 0.547i)7-s − 0.738i·11-s + (0.554 + 0.554i)13-s + (0.109 − 0.109i)17-s + 0.458·19-s + (0.0937 − 0.0937i)23-s + (0.979 + 0.200i)25-s + 0.0834i·29-s − 1.59i·31-s + (−0.490 − 0.600i)35-s + (0.476 − 0.476i)37-s − 0.765·41-s + (0.914 − 0.914i)43-s + (1.36 + 1.36i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.945270739\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.945270739\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.22 - 0.224i)T \) |
good | 7 | \( 1 + (1.44 + 1.44i)T + 7iT^{2} \) |
| 11 | \( 1 + 2.44iT - 11T^{2} \) |
| 13 | \( 1 + (-2 - 2i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.449 + 0.449i)T - 17iT^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + (-0.449 + 0.449i)T - 23iT^{2} \) |
| 29 | \( 1 - 0.449iT - 29T^{2} \) |
| 31 | \( 1 + 8.89iT - 31T^{2} \) |
| 37 | \( 1 + (-2.89 + 2.89i)T - 37iT^{2} \) |
| 41 | \( 1 + 4.89T + 41T^{2} \) |
| 43 | \( 1 + (-6 + 6i)T - 43iT^{2} \) |
| 47 | \( 1 + (-9.34 - 9.34i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.89 + 2.89i)T + 53iT^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + (6.89 + 6.89i)T + 67iT^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 + (-7.89 - 7.89i)T + 73iT^{2} \) |
| 79 | \( 1 - 3.10T + 79T^{2} \) |
| 83 | \( 1 + (-5.55 + 5.55i)T - 83iT^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 + (-3 + 3i)T - 97iT^{2} \) |
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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
−9.427184882276224851748554723952, −8.876666988737956904459531191906, −7.78719881351750329681256838353, −6.89275401297866063594314921842, −6.12176940237948063943271081238, −5.52433586416336050860103365784, −4.27313689992625560565751424729, −3.33479139889774377640056037894, −2.24921700038443239087836542337, −0.889145791518143419807861616668,
1.29664129240195812460883137897, 2.48783365739731674744688644026, 3.38998894656547614869955172195, 4.74836876970103008867975065686, 5.54756580737387718590019593549, 6.26434153043606672627995506095, 7.04831298592213215923383339576, 8.101922894053295689050133441282, 9.014260625929050362142769213135, 9.538332899733149268306095533487