L(s) = 1 | + (−3.28e3 − 5.68e3i)3-s + (−3.27e4 − 5.67e4i)4-s + (−5.73e6 − 5.73e5i)7-s + (−2.15e7 + 3.72e7i)9-s + (−2.14e8 + 3.72e8i)12-s − 1.20e9·13-s + (−2.14e9 + 3.71e9i)16-s + (1.19e10 − 2.06e10i)19-s + (1.55e10 + 3.44e10i)21-s + (−7.62e10 − 1.32e11i)25-s + 2.82e11·27-s + (1.55e11 + 3.44e11i)28-s + (1.01e11 + 1.75e11i)31-s + 2.82e12·36-s + (−8.83e11 + 1.52e12i)37-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (−0.995 − 0.0994i)7-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)12-s − 1.47·13-s + (−0.499 + 0.866i)16-s + (0.702 − 1.21i)19-s + (0.411 + 0.911i)21-s + (−0.5 − 0.866i)25-s + 27-s + (0.411 + 0.911i)28-s + (0.118 + 0.205i)31-s + 0.999·36-s + (−0.251 + 0.435i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{17}{2})\) |
\(\approx\) |
\(0.3121798390\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3121798390\) |
\(L(9)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (3.28e3 + 5.68e3i)T \) |
| 7 | \( 1 + (5.73e6 + 5.73e5i)T \) |
good | 2 | \( 1 + (3.27e4 + 5.67e4i)T^{2} \) |
| 5 | \( 1 + (7.62e10 + 1.32e11i)T^{2} \) |
| 11 | \( 1 + (2.29e16 - 3.97e16i)T^{2} \) |
| 13 | \( 1 + 1.20e9T + 6.65e17T^{2} \) |
| 17 | \( 1 + (2.43e19 - 4.21e19i)T^{2} \) |
| 19 | \( 1 + (-1.19e10 + 2.06e10i)T + (-1.44e20 - 2.49e20i)T^{2} \) |
| 23 | \( 1 + (3.06e21 + 5.31e21i)T^{2} \) |
| 29 | \( 1 - 2.50e23T^{2} \) |
| 31 | \( 1 + (-1.01e11 - 1.75e11i)T + (-3.63e23 + 6.29e23i)T^{2} \) |
| 37 | \( 1 + (8.83e11 - 1.52e12i)T + (-6.16e24 - 1.06e25i)T^{2} \) |
| 41 | \( 1 - 6.37e25T^{2} \) |
| 43 | \( 1 - 2.33e13T + 1.36e26T^{2} \) |
| 47 | \( 1 + (2.83e26 + 4.91e26i)T^{2} \) |
| 53 | \( 1 + (1.93e27 - 3.35e27i)T^{2} \) |
| 59 | \( 1 + (1.07e28 - 1.86e28i)T^{2} \) |
| 61 | \( 1 + (9.21e13 - 1.59e14i)T + (-1.83e28 - 3.18e28i)T^{2} \) |
| 67 | \( 1 + (2.89e14 + 5.01e14i)T + (-8.24e28 + 1.42e29i)T^{2} \) |
| 71 | \( 1 - 4.16e29T^{2} \) |
| 73 | \( 1 + (7.19e14 + 1.24e15i)T + (-3.25e29 + 5.63e29i)T^{2} \) |
| 79 | \( 1 + (1.28e15 - 2.22e15i)T + (-1.15e30 - 1.99e30i)T^{2} \) |
| 83 | \( 1 - 5.07e30T^{2} \) |
| 89 | \( 1 + (7.74e30 + 1.34e31i)T^{2} \) |
| 97 | \( 1 - 1.56e16T + 6.14e31T^{2} \) |
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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
−14.20870593026171018778816415169, −13.20280165745625455378174840976, −12.02861362738924948912521611023, −10.44475188677537248769035349558, −9.259947904168653561591912381712, −7.31807551412554410797416230302, −6.11375274074561265591952069944, −4.82053383880090644487836296140, −2.49034723332367872108629798395, −0.77192071892843547020338994618,
0.14919050651065554542519296887, 2.97968316825045336328788213138, 4.13967408207689331256391446081, 5.60520130029096884081248828227, 7.40559094551865126543128498029, 9.187543426116946574710320880828, 10.03118491712803341510074065792, 11.83676886597536103284025073196, 12.72050149621963080405138880711, 14.33456021006174895443124797327