L(s) = 1 | + (0.5 + 0.866i)3-s + (−4.5 − 2.59i)5-s + (1 + 6.92i)7-s + (4 − 6.92i)9-s + (8.5 + 14.7i)11-s + 13.8i·13-s − 5.19i·15-s + (12.5 + 21.6i)17-s + (−3.5 + 6.06i)19-s + (−5.49 + 4.33i)21-s + (−4.5 − 2.59i)23-s + (1 + 1.73i)25-s + 17·27-s − 13.8i·29-s + (−28.5 + 16.4i)31-s + ⋯ |
L(s) = 1 | + (0.166 + 0.288i)3-s + (−0.900 − 0.519i)5-s + (0.142 + 0.989i)7-s + (0.444 − 0.769i)9-s + (0.772 + 1.33i)11-s + 1.06i·13-s − 0.346i·15-s + (0.735 + 1.27i)17-s + (−0.184 + 0.319i)19-s + (−0.261 + 0.206i)21-s + (−0.195 − 0.112i)23-s + (0.0400 + 0.0692i)25-s + 0.629·27-s − 0.477i·29-s + (−0.919 + 0.530i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.12548 + 0.815380i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12548 + 0.815380i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-1 - 6.92i)T \) |
good | 3 | \( 1 + (-0.5 - 0.866i)T + (-4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (4.5 + 2.59i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-8.5 - 14.7i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 13.8iT - 169T^{2} \) |
| 17 | \( 1 + (-12.5 - 21.6i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (3.5 - 6.06i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (4.5 + 2.59i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 13.8iT - 841T^{2} \) |
| 31 | \( 1 + (28.5 - 16.4i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-7.5 - 4.33i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 26T + 1.68e3T^{2} \) |
| 43 | \( 1 + 14T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-43.5 - 25.1i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-79.5 + 45.8i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (27.5 + 47.6i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-19.5 - 11.2i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-8.5 - 14.7i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + (59.5 + 103. i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (64.5 + 37.2i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 110T + 6.88e3T^{2} \) |
| 89 | \( 1 + (35.5 - 61.4i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 22T + 9.40e3T^{2} \) |
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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
−12.25483576392990366429238027063, −11.59329595281554338129202678429, −10.07795307176258311849264804878, −9.212905316801777942365979451242, −8.449809937801265543653716149277, −7.23933598184862773709201889110, −6.07851328555039675789654090660, −4.50761519667970612628382964577, −3.81447729561039505965949254394, −1.74670920262130021193647966917,
0.815849817403449677751324828106, 3.05667365462859865358890721342, 4.09144513729165110344838127044, 5.58996298332999203254156216581, 7.19416498957755656186953152040, 7.56737398640980444868945323185, 8.662716345705989752133331530599, 10.11402219783600085443445947050, 10.99379657716838765626466411014, 11.60301348168794433735797890355