| L(s) = 1 | − 4·2-s + 26.7·3-s + 16·4-s + 50.5·5-s − 106.·6-s + 248.·7-s − 64·8-s + 472.·9-s − 202.·10-s + 32.9·11-s + 427.·12-s − 501.·13-s − 992.·14-s + 1.35e3·15-s + 256·16-s + 658.·17-s − 1.89e3·18-s − 1.91e3·19-s + 808.·20-s + 6.63e3·21-s − 131.·22-s + 3.25e3·23-s − 1.71e3·24-s − 568.·25-s + 2.00e3·26-s + 6.13e3·27-s + 3.96e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.71·3-s + 0.5·4-s + 0.904·5-s − 1.21·6-s + 1.91·7-s − 0.353·8-s + 1.94·9-s − 0.639·10-s + 0.0820·11-s + 0.857·12-s − 0.823·13-s − 1.35·14-s + 1.55·15-s + 0.250·16-s + 0.552·17-s − 1.37·18-s − 1.21·19-s + 0.452·20-s + 3.28·21-s − 0.0580·22-s + 1.28·23-s − 0.606·24-s − 0.182·25-s + 0.582·26-s + 1.62·27-s + 0.956·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.783885613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.783885613\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 269 | \( 1 - 7.23e4T \) |
| good | 3 | \( 1 - 26.7T + 243T^{2} \) |
| 5 | \( 1 - 50.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 248.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 32.9T + 1.61e5T^{2} \) |
| 13 | \( 1 + 501.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 658.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.91e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.25e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.07e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.74e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.57e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.82e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 804.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.95e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 813.T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.16e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.22e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 8.72e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.29e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.36e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.76e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.13e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.28e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.42e5T + 8.58e9T^{2} \) |
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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
−9.765898587512248261852968454545, −9.066293175822729396164605924426, −8.235867854888496458651746233009, −7.83122685348487346051104550558, −6.81073433935763179591345874819, −5.27931742291873261486697931665, −4.25073725091371872303906100888, −2.71946414533525483721781732457, −2.01669771161414398101380547359, −1.26963555233854712671910749804,
1.26963555233854712671910749804, 2.01669771161414398101380547359, 2.71946414533525483721781732457, 4.25073725091371872303906100888, 5.27931742291873261486697931665, 6.81073433935763179591345874819, 7.83122685348487346051104550558, 8.235867854888496458651746233009, 9.066293175822729396164605924426, 9.765898587512248261852968454545
