| L(s) = 1 | + 3.01·2-s − 13.7·3-s − 22.9·4-s − 25·5-s − 41.3·6-s + 131.·7-s − 165.·8-s − 54.9·9-s − 75.3·10-s + 314.·12-s − 598.·13-s + 396.·14-s + 342.·15-s + 234.·16-s + 1.33e3·17-s − 165.·18-s − 2.88e3·19-s + 572.·20-s − 1.80e3·21-s − 2.04e3·23-s + 2.27e3·24-s + 625·25-s − 1.80e3·26-s + 4.08e3·27-s − 3.01e3·28-s − 5.80e3·29-s + 1.03e3·30-s + ⋯ |
| L(s) = 1 | + 0.532·2-s − 0.879·3-s − 0.716·4-s − 0.447·5-s − 0.468·6-s + 1.01·7-s − 0.914·8-s − 0.225·9-s − 0.238·10-s + 0.630·12-s − 0.982·13-s + 0.540·14-s + 0.393·15-s + 0.228·16-s + 1.12·17-s − 0.120·18-s − 1.83·19-s + 0.320·20-s − 0.892·21-s − 0.807·23-s + 0.804·24-s + 0.200·25-s − 0.523·26-s + 1.07·27-s − 0.726·28-s − 1.28·29-s + 0.209·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4848284751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4848284751\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 3.01T + 32T^{2} \) |
| 3 | \( 1 + 13.7T + 243T^{2} \) |
| 7 | \( 1 - 131.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 598.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.33e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.88e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.04e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.80e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.52e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.18e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.34e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.52e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.17e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.32e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.36e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.64e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.83e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.71e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.34e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.87e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 350.T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.59e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.53e4T + 8.58e9T^{2} \) |
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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
−10.06345582713627941739378493241, −8.864762932971198065759170333909, −8.166730817337031499871817977929, −7.15811057647715267892173068864, −5.84265715292339421962964081885, −5.26966567145496628979802512179, −4.47337868458041105818278436309, −3.52972737341713640022144957694, −1.94214815947172741618376181519, −0.31625859491327427867674244216,
0.31625859491327427867674244216, 1.94214815947172741618376181519, 3.52972737341713640022144957694, 4.47337868458041105818278436309, 5.26966567145496628979802512179, 5.84265715292339421962964081885, 7.15811057647715267892173068864, 8.166730817337031499871817977929, 8.864762932971198065759170333909, 10.06345582713627941739378493241
