L(s) = 1 | + 2.46·2-s + 4.05·4-s − 3.65·5-s + 5.05·8-s − 9.00·10-s + 0.406·11-s − 0.486·13-s + 4.32·16-s + 4.85·17-s − 1.97·19-s − 14.8·20-s + 22-s + 4.64·23-s + 8.38·25-s − 1.19·26-s + 7.64·29-s + 7.02·31-s + 0.539·32-s + 11.9·34-s + 2.32·37-s − 4.85·38-s − 18.4·40-s + 7.51·41-s − 2.32·43-s + 1.64·44-s + 11.4·46-s − 6.31·47-s + ⋯ |
L(s) = 1 | + 1.73·2-s + 2.02·4-s − 1.63·5-s + 1.78·8-s − 2.84·10-s + 0.122·11-s − 0.135·13-s + 1.08·16-s + 1.17·17-s − 0.452·19-s − 3.31·20-s + 0.213·22-s + 0.969·23-s + 1.67·25-s − 0.234·26-s + 1.42·29-s + 1.26·31-s + 0.0953·32-s + 2.04·34-s + 0.382·37-s − 0.787·38-s − 2.92·40-s + 1.17·41-s − 0.354·43-s + 0.248·44-s + 1.68·46-s − 0.921·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.358922863\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.358922863\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2.46T + 2T^{2} \) |
| 5 | \( 1 + 3.65T + 5T^{2} \) |
| 11 | \( 1 - 0.406T + 11T^{2} \) |
| 13 | \( 1 + 0.486T + 13T^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 19 | \( 1 + 1.97T + 19T^{2} \) |
| 23 | \( 1 - 4.64T + 23T^{2} \) |
| 29 | \( 1 - 7.64T + 29T^{2} \) |
| 31 | \( 1 - 7.02T + 31T^{2} \) |
| 37 | \( 1 - 2.32T + 37T^{2} \) |
| 41 | \( 1 - 7.51T + 41T^{2} \) |
| 43 | \( 1 + 2.32T + 43T^{2} \) |
| 47 | \( 1 + 6.31T + 47T^{2} \) |
| 53 | \( 1 + 3.56T + 53T^{2} \) |
| 59 | \( 1 - 6.11T + 59T^{2} \) |
| 61 | \( 1 + 8.02T + 61T^{2} \) |
| 67 | \( 1 - 3.60T + 67T^{2} \) |
| 71 | \( 1 - 8.46T + 71T^{2} \) |
| 73 | \( 1 - 1.97T + 73T^{2} \) |
| 79 | \( 1 - 8.16T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 + 9.48T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.022762042478479235695171006235, −7.67543836017403036812015061090, −6.71783205916777896612598661553, −6.26954518680272875440739835469, −5.07044247816398670433938369953, −4.69790832795051415766745486478, −3.88181932460863600953722406991, −3.28709052250769518220578943218, −2.57474905930389913073495002766, −0.943066418682506317245108975028,
0.943066418682506317245108975028, 2.57474905930389913073495002766, 3.28709052250769518220578943218, 3.88181932460863600953722406991, 4.69790832795051415766745486478, 5.07044247816398670433938369953, 6.26954518680272875440739835469, 6.71783205916777896612598661553, 7.67543836017403036812015061090, 8.022762042478479235695171006235