L(s) = 1 | − 2.36·2-s − 3·3-s − 2.42·4-s − 6.42·5-s + 7.08·6-s + 29.4·7-s + 24.6·8-s + 9·9-s + 15.1·10-s + 0.624·11-s + 7.26·12-s − 69.6·14-s + 19.2·15-s − 38.7·16-s + 87.7·17-s − 21.2·18-s − 82.8·19-s + 15.5·20-s − 88.4·21-s − 1.47·22-s − 74.7·23-s − 73.8·24-s − 83.7·25-s − 27·27-s − 71.4·28-s + 226.·29-s − 45.5·30-s + ⋯ |
L(s) = 1 | − 0.835·2-s − 0.577·3-s − 0.302·4-s − 0.574·5-s + 0.482·6-s + 1.59·7-s + 1.08·8-s + 0.333·9-s + 0.479·10-s + 0.0171·11-s + 0.174·12-s − 1.32·14-s + 0.331·15-s − 0.605·16-s + 1.25·17-s − 0.278·18-s − 0.999·19-s + 0.173·20-s − 0.919·21-s − 0.0142·22-s − 0.678·23-s − 0.628·24-s − 0.670·25-s − 0.192·27-s − 0.482·28-s + 1.44·29-s − 0.276·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8769066822\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8769066822\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2.36T + 8T^{2} \) |
| 5 | \( 1 + 6.42T + 125T^{2} \) |
| 7 | \( 1 - 29.4T + 343T^{2} \) |
| 11 | \( 1 - 0.624T + 1.33e3T^{2} \) |
| 17 | \( 1 - 87.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 82.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 74.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 226.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 173.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 112.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 267.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 383.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 337.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 146.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 529.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 203.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 121.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 661.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 167.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 101.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 506.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.40e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.90e3T + 9.12e5T^{2} \) |
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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
−10.54475776489834784152005792517, −9.676039661550176373003845255371, −8.463491136161441194477471211524, −8.016913113431505876496606466256, −7.23929082655130372529532620872, −5.73363883970747809119022688912, −4.75199752650937116536138547260, −3.98940217673127244186006041620, −1.84128137845546088408158905928, −0.71098487746157144149232049275,
0.71098487746157144149232049275, 1.84128137845546088408158905928, 3.98940217673127244186006041620, 4.75199752650937116536138547260, 5.73363883970747809119022688912, 7.23929082655130372529532620872, 8.016913113431505876496606466256, 8.463491136161441194477471211524, 9.676039661550176373003845255371, 10.54475776489834784152005792517