L(s) = 1 | − 2.37·5-s + 2.52i·7-s − 2.52i·11-s − 1.58i·13-s + 0.372·17-s + (4 + 1.73i)19-s − 1.87i·23-s + 0.627·25-s − 3.16i·29-s − 2.74·31-s − 5.98i·35-s − 1.58i·37-s + 6.92i·41-s − 0.644i·43-s − 0.939i·47-s + ⋯ |
L(s) = 1 | − 1.06·5-s + 0.954i·7-s − 0.761i·11-s − 0.439i·13-s + 0.0902·17-s + (0.917 + 0.397i)19-s − 0.391i·23-s + 0.125·25-s − 0.588i·29-s − 0.492·31-s − 1.01i·35-s − 0.260i·37-s + 1.08i·41-s − 0.0983i·43-s − 0.137i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.277426738\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.277426738\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-4 - 1.73i)T \) |
good | 5 | \( 1 + 2.37T + 5T^{2} \) |
| 7 | \( 1 - 2.52iT - 7T^{2} \) |
| 11 | \( 1 + 2.52iT - 11T^{2} \) |
| 13 | \( 1 + 1.58iT - 13T^{2} \) |
| 17 | \( 1 - 0.372T + 17T^{2} \) |
| 23 | \( 1 + 1.87iT - 23T^{2} \) |
| 29 | \( 1 + 3.16iT - 29T^{2} \) |
| 31 | \( 1 + 2.74T + 31T^{2} \) |
| 37 | \( 1 + 1.58iT - 37T^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 0.644iT - 43T^{2} \) |
| 47 | \( 1 + 0.939iT - 47T^{2} \) |
| 53 | \( 1 - 10.0iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 0.372T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 6.74T + 79T^{2} \) |
| 83 | \( 1 + 3.46iT - 83T^{2} \) |
| 89 | \( 1 - 13.2iT - 89T^{2} \) |
| 97 | \( 1 - 13.2iT - 97T^{2} \) |
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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
−8.811150922411199207900730979173, −8.035786314457096910329453818648, −7.67967002013739861012501136935, −6.57884704225468047444702681294, −5.75324240736233862155050319588, −5.11375607271090113218070796303, −3.99496771459034468944252521091, −3.29681036990963501770902268408, −2.37276242559916691270723132503, −0.825027235551973116206661704269,
0.59870720590271626844612130626, 1.90136343353469767999038587950, 3.31217979086211243511308834387, 3.93212728948566397945000114991, 4.68324467876313276239246987529, 5.52340269597874660299111335003, 6.88443820252751587507008177690, 7.16760835549502280031383844642, 7.85453940859785138146928638183, 8.649827278102811512429313810163