L(s) = 1 | − 4.47·2-s − 11.9·4-s − 224.·7-s + 196.·8-s − 501.·11-s − 224.·13-s + 1.00e3·14-s − 496.·16-s − 1.66e3·17-s + 484·19-s + 2.24e3·22-s − 2.26e3·23-s + 1.00e3·26-s + 2.69e3·28-s − 5.52e3·29-s + 3.60e3·31-s − 4.07e3·32-s + 7.46e3·34-s − 7.40e3·37-s − 2.16e3·38-s + 1.10e4·41-s − 1.25e4·43-s + 6.02e3·44-s + 1.01e4·46-s − 9.54e3·47-s + 3.35e4·49-s + 2.69e3·52-s + ⋯ |
L(s) = 1 | − 0.790·2-s − 0.374·4-s − 1.73·7-s + 1.08·8-s − 1.25·11-s − 0.368·13-s + 1.36·14-s − 0.484·16-s − 1.39·17-s + 0.307·19-s + 0.988·22-s − 0.891·23-s + 0.291·26-s + 0.649·28-s − 1.21·29-s + 0.674·31-s − 0.704·32-s + 1.10·34-s − 0.889·37-s − 0.243·38-s + 1.02·41-s − 1.03·43-s + 0.469·44-s + 0.705·46-s − 0.630·47-s + 1.99·49-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.2160585139\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2160585139\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 4.47T + 32T^{2} \) |
| 7 | \( 1 + 224.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 501.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 224.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.66e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 484T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.26e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.52e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.60e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.40e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.10e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.25e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.54e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.77e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.52e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.13e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.45e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.31e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.69e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.96e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.71e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.20e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.42e4T + 8.58e9T^{2} \) |
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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
−11.04628532255598988332633466799, −10.05725409811950433603317381405, −9.573931262540467923243932479084, −8.564728362099047094053601344812, −7.50573997781794297036214276731, −6.46657420796636436536923048516, −5.12969983867077137146956585999, −3.74542760636028952782164098529, −2.33386683535418631157816049537, −0.29539854319508773744557523772,
0.29539854319508773744557523772, 2.33386683535418631157816049537, 3.74542760636028952782164098529, 5.12969983867077137146956585999, 6.46657420796636436536923048516, 7.50573997781794297036214276731, 8.564728362099047094053601344812, 9.573931262540467923243932479084, 10.05725409811950433603317381405, 11.04628532255598988332633466799