L(s) = 1 | + 1.19i·2-s + (0.625 − 2.93i)3-s + 2.57·4-s − 0.192i·5-s + (3.49 + 0.745i)6-s + 6.00·7-s + 7.84i·8-s + (−8.21 − 3.66i)9-s + 0.229·10-s − 7.32i·11-s + (1.61 − 7.56i)12-s + 9.24·13-s + 7.15i·14-s + (−0.565 − 0.120i)15-s + 0.952·16-s − 9.92i·17-s + ⋯ |
L(s) = 1 | + 0.596i·2-s + (0.208 − 0.978i)3-s + 0.644·4-s − 0.0385i·5-s + (0.583 + 0.124i)6-s + 0.857·7-s + 0.980i·8-s + (−0.913 − 0.407i)9-s + 0.0229·10-s − 0.666i·11-s + (0.134 − 0.630i)12-s + 0.710·13-s + 0.511i·14-s + (−0.0376 − 0.00803i)15-s + 0.0595·16-s − 0.583i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.98472 - 0.209073i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.98472 - 0.209073i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.625 + 2.93i)T \) |
| 59 | \( 1 + 7.68iT \) |
good | 2 | \( 1 - 1.19iT - 4T^{2} \) |
| 5 | \( 1 + 0.192iT - 25T^{2} \) |
| 7 | \( 1 - 6.00T + 49T^{2} \) |
| 11 | \( 1 + 7.32iT - 121T^{2} \) |
| 13 | \( 1 - 9.24T + 169T^{2} \) |
| 17 | \( 1 + 9.92iT - 289T^{2} \) |
| 19 | \( 1 - 0.266T + 361T^{2} \) |
| 23 | \( 1 + 9.30iT - 529T^{2} \) |
| 29 | \( 1 - 28.6iT - 841T^{2} \) |
| 31 | \( 1 + 36.4T + 961T^{2} \) |
| 37 | \( 1 - 9.90T + 1.36e3T^{2} \) |
| 41 | \( 1 - 38.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 39.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 17.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 23.3iT - 2.80e3T^{2} \) |
| 61 | \( 1 + 71.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 58.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 25.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 68.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 72.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 85.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 94.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 150.T + 9.40e3T^{2} \) |
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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
−12.42734081639217668002478145486, −11.40014628945246386316765379566, −10.86717578174422997815893213564, −8.873281393295430553324467992263, −8.154050322175170946025002516314, −7.21212401940420339162825824550, −6.28900825343066237533960135685, −5.20577420431902506605405198682, −3.01295140450173346460469130173, −1.48027953298318755065302542307,
1.87293223188986844386301620149, 3.38610009889097709897150242225, 4.56296543038175641832257777421, 5.95367564699274296524839478186, 7.44175445716549536231010765044, 8.605714253826153048809458221918, 9.744684092766992133413201601412, 10.69580252220115429986535160179, 11.23404338739231408569380416778, 12.20877362293243902178463401030