L(s) = 1 | − 1.07i·2-s + 5.19·3-s + 14.8·4-s + 26.1·5-s − 5.60i·6-s + 49.1·7-s − 33.2i·8-s + 27·9-s − 28.1i·10-s − 190. i·11-s + 77.1·12-s + 198. i·13-s − 52.9i·14-s + 135.·15-s + 201.·16-s − 81.6·17-s + ⋯ |
L(s) = 1 | − 0.269i·2-s + 0.577·3-s + 0.927·4-s + 1.04·5-s − 0.155i·6-s + 1.00·7-s − 0.519i·8-s + 0.333·9-s − 0.281i·10-s − 1.57i·11-s + 0.535·12-s + 1.17i·13-s − 0.270i·14-s + 0.603·15-s + 0.787·16-s − 0.282·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(3.625097332\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.625097332\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 5.19T \) |
| 59 | \( 1 + (3.01e3 + 1.73e3i)T \) |
good | 2 | \( 1 + 1.07iT - 16T^{2} \) |
| 5 | \( 1 - 26.1T + 625T^{2} \) |
| 7 | \( 1 - 49.1T + 2.40e3T^{2} \) |
| 11 | \( 1 + 190. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 198. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 81.6T + 8.35e4T^{2} \) |
| 19 | \( 1 + 590.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 808. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 714.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.37e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 2.10e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.13e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 349. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.25e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 4.73e3T + 7.89e6T^{2} \) |
| 61 | \( 1 + 1.50e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 233. iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.66e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 9.12e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 8.28e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 7.47e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.05e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.50e4iT - 8.85e7T^{2} \) |
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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
−11.54231806372425622935306819372, −11.16458817043536273288676534692, −9.970629855433907439204574008736, −8.909609845150393104874211900428, −7.917130946834949753874673317050, −6.58254849682825496199785153086, −5.66599312989097717050117810705, −3.90907018775507364427667517538, −2.35141874523317527858375546053, −1.53832079695837321375646877806,
1.79220830373544717316347366631, 2.45084226348020311308212278317, 4.53190175389419617862373618726, 5.78565094697021556822777578425, 6.95404594863031986634976564172, 7.891709922467858209473814073572, 8.941071318423947646256667578570, 10.33135201350625146198255620921, 10.74666692890276938892965208439, 12.34579329463157173551241664990