L(s) = 1 | − 0.745·2-s + 3.05·3-s − 1.44·4-s + 5-s − 2.28·6-s − 3.66·7-s + 2.56·8-s + 6.35·9-s − 0.745·10-s + 11-s − 4.41·12-s + 1.02·13-s + 2.72·14-s + 3.05·15-s + 0.974·16-s + 7.92·17-s − 4.73·18-s − 1.53·19-s − 1.44·20-s − 11.1·21-s − 0.745·22-s − 9.02·23-s + 7.85·24-s + 25-s − 0.764·26-s + 10.2·27-s + 5.28·28-s + ⋯ |
L(s) = 1 | − 0.527·2-s + 1.76·3-s − 0.722·4-s + 0.447·5-s − 0.930·6-s − 1.38·7-s + 0.907·8-s + 2.11·9-s − 0.235·10-s + 0.301·11-s − 1.27·12-s + 0.284·13-s + 0.729·14-s + 0.789·15-s + 0.243·16-s + 1.92·17-s − 1.11·18-s − 0.352·19-s − 0.322·20-s − 2.44·21-s − 0.158·22-s − 1.88·23-s + 1.60·24-s + 0.200·25-s − 0.150·26-s + 1.97·27-s + 0.999·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.384066780\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.384066780\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
good | 2 | \( 1 + 0.745T + 2T^{2} \) |
| 3 | \( 1 - 3.05T + 3T^{2} \) |
| 7 | \( 1 + 3.66T + 7T^{2} \) |
| 13 | \( 1 - 1.02T + 13T^{2} \) |
| 17 | \( 1 - 7.92T + 17T^{2} \) |
| 19 | \( 1 + 1.53T + 19T^{2} \) |
| 23 | \( 1 + 9.02T + 23T^{2} \) |
| 29 | \( 1 - 3.19T + 29T^{2} \) |
| 31 | \( 1 + 7.54T + 31T^{2} \) |
| 37 | \( 1 - 6.07T + 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 43 | \( 1 - 8.29T + 43T^{2} \) |
| 47 | \( 1 - 9.63T + 47T^{2} \) |
| 53 | \( 1 + 3.09T + 53T^{2} \) |
| 59 | \( 1 - 8.32T + 59T^{2} \) |
| 61 | \( 1 + 7.95T + 61T^{2} \) |
| 67 | \( 1 + 2.85T + 67T^{2} \) |
| 71 | \( 1 + 3.17T + 71T^{2} \) |
| 79 | \( 1 - 16.2T + 79T^{2} \) |
| 83 | \( 1 + 1.77T + 83T^{2} \) |
| 89 | \( 1 - 0.725T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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
−8.612585578718715894993640592402, −7.72455959780799364051325109704, −7.53651111392356940892564310789, −6.29424943013772607204356225553, −5.61691784468099080426598967384, −4.18118509846470484430668375705, −3.76553584267362811256223280931, −2.98954481578456349130131559983, −2.04315561318937665694717649289, −0.914762605455101139972955805406,
0.914762605455101139972955805406, 2.04315561318937665694717649289, 2.98954481578456349130131559983, 3.76553584267362811256223280931, 4.18118509846470484430668375705, 5.61691784468099080426598967384, 6.29424943013772607204356225553, 7.53651111392356940892564310789, 7.72455959780799364051325109704, 8.612585578718715894993640592402