L(s) = 1 | + 7.07i·5-s − 60·7-s − 333. i·11-s + 1.19e3·13-s + 4.15e3i·17-s − 8.43e3·19-s − 6.40e3i·23-s + 1.55e4·25-s + 3.36e4i·29-s − 4.68e4·31-s − 424. i·35-s + 1.09e4·37-s + 7.34e4i·41-s − 5.94e4·43-s + 1.17e5i·47-s + ⋯ |
L(s) = 1 | + 0.0565i·5-s − 0.174·7-s − 0.250i·11-s + 0.542·13-s + 0.844i·17-s − 1.22·19-s − 0.526i·23-s + 0.996·25-s + 1.37i·29-s − 1.57·31-s − 0.00989i·35-s + 0.215·37-s + 1.06i·41-s − 0.747·43-s + 1.13i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.9136458261\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9136458261\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 7.07iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 60T + 1.17e5T^{2} \) |
| 11 | \( 1 + 333. iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 1.19e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 4.15e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 8.43e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 6.40e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 3.36e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 4.68e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 1.09e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 7.34e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 5.94e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.17e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 8.23e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 2.81e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 3.39e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 1.48e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 4.11e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 4.01e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 7.91e4T + 2.43e11T^{2} \) |
| 83 | \( 1 - 1.00e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 3.75e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 6.63e5T + 8.32e11T^{2} \) |
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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
−12.54927056603948744933109554740, −11.09324512183712006405123540529, −10.49908771264549588716998769016, −9.089522566700895296861736595230, −8.273610363282247919269507260377, −6.87521545485434035972625884507, −5.89038772865405389154305606633, −4.42893098848402610733113065341, −3.11253819860446340463230459126, −1.47910202474203878469037940314,
0.27225694244022095820349282128, 1.98871286830592247055630855008, 3.53720866931624562540561213301, 4.87809207604277755779888416324, 6.19458344039113520180978268508, 7.30036529474617142676494404217, 8.538730003652939580382319980357, 9.511875543277802025159704091880, 10.64279936569590507720808104100, 11.58301543043694879858240903618