L(s) = 1 | + (1 + 1.73i)2-s + (−1.5 + 2.59i)3-s + (−1.99 + 3.46i)4-s + (3 + 5.19i)5-s − 6·6-s − 7.99·8-s + (−4.5 − 7.79i)9-s + (−6 + 10.3i)10-s + (−6 + 10.3i)11-s + (−6.00 − 10.3i)12-s − 38·13-s − 18·15-s + (−8 − 13.8i)16-s + (−63 + 109. i)17-s + (9 − 15.5i)18-s + (10 + 17.3i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.268 + 0.464i)5-s − 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.189 + 0.328i)10-s + (−0.164 + 0.284i)11-s + (−0.144 − 0.249i)12-s − 0.810·13-s − 0.309·15-s + (−0.125 − 0.216i)16-s + (−0.898 + 1.55i)17-s + (0.117 − 0.204i)18-s + (0.120 + 0.209i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.7518815567\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7518815567\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 - 1.73i)T \) |
| 3 | \( 1 + (1.5 - 2.59i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-3 - 5.19i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (6 - 10.3i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 38T + 2.19e3T^{2} \) |
| 17 | \( 1 + (63 - 109. i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-10 - 17.3i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (84 + 145. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 30T + 2.43e4T^{2} \) |
| 31 | \( 1 + (44 - 76.2i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (127 + 219. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 42T + 6.89e4T^{2} \) |
| 43 | \( 1 + 52T + 7.95e4T^{2} \) |
| 47 | \( 1 + (48 + 83.1i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (99 - 171. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (330 - 571. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (269 + 465. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (442 - 765. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 792T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-109 + 188. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-260 - 450. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 492T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-405 - 701. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.15e3T + 9.12e5T^{2} \) |
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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
−12.15216547835482556883230330950, −10.74544395123910421163138073958, −10.23664866109193176100804409578, −9.017334210720273120144212115221, −8.043162932754414211602996240800, −6.79539700521427949577129819070, −6.07233260066786283124787533782, −4.87102919833459958809005499664, −3.91458159202108903596718026235, −2.36501203837609088813229901009,
0.24399738006209510674880609782, 1.76082689393026241595696109359, 3.05458293498778730092072136397, 4.72013885712933269044801404359, 5.44477504019580419359726111075, 6.72455575328038498838498487116, 7.78120649299096865921239286355, 9.115861657262346682762043039465, 9.790468505158552916407266786254, 11.05662783862247437044188632211