L(s) = 1 | + (1.90 + 1.09i)2-s + (5.19 − 0.206i)3-s + (−1.59 − 2.75i)4-s + (10.0 + 5.30i)6-s + (−2.39 − 1.38i)7-s − 24.5i·8-s + (26.9 − 2.14i)9-s + (26.3 − 45.6i)11-s + (−8.83 − 13.9i)12-s + (−17.7 + 10.2i)13-s + (−3.03 − 5.25i)14-s + (14.1 − 24.5i)16-s + 3.66i·17-s + (53.4 + 25.4i)18-s + 95.6·19-s + ⋯ |
L(s) = 1 | + (0.671 + 0.387i)2-s + (0.999 − 0.0396i)3-s + (−0.199 − 0.344i)4-s + (0.686 + 0.360i)6-s + (−0.129 − 0.0746i)7-s − 1.08i·8-s + (0.996 − 0.0792i)9-s + (0.721 − 1.25i)11-s + (−0.212 − 0.336i)12-s + (−0.377 + 0.218i)13-s + (−0.0579 − 0.100i)14-s + (0.221 − 0.384i)16-s + 0.0522i·17-s + (0.700 + 0.333i)18-s + 1.15·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.19219 - 0.915884i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.19219 - 0.915884i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-5.19 + 0.206i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-1.90 - 1.09i)T + (4 + 6.92i)T^{2} \) |
| 7 | \( 1 + (2.39 + 1.38i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-26.3 + 45.6i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (17.7 - 10.2i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 3.66iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 95.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-77.8 + 44.9i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (113. - 197. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (139. + 241. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 273. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + (32.4 + 56.1i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-362. - 209. i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (120. + 69.3i)T + (5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 197. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-370. - 641. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (244. - 423. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (356. - 205. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 310.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 51.0iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-603. + 1.04e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-783. - 452. i)T + (2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 663.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-628. - 362. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.92163861205592549426486054920, −10.64785670294522819984146085878, −9.472570293908711374030195020533, −8.935409455719914962599970027450, −7.56302790340225630661402856590, −6.58244565646261407960496166598, −5.40900729117598996562589365503, −4.10782986143302440573055061942, −3.12870343309761851678505234725, −1.13482350333067897169575884894,
1.92083530208652359295705332278, 3.17445379350757495227676493230, 4.14734291503442939293996734246, 5.22057153801901106427638790803, 7.09844095568644589789971998175, 7.83256191946683907849924935154, 9.131891291831141601612036567457, 9.666457584981060055627875293016, 11.10649516233927395959193180822, 12.29472440969651754448208658055