| L(s) = 1 | + (0.989 − 0.130i)2-s + (0.257 − 1.71i)3-s + (−0.970 + 0.259i)4-s + (−1.66 − 3.37i)5-s + (0.0316 − 1.72i)6-s + (3.57 + 1.76i)7-s + (−2.76 + 1.14i)8-s + (−2.86 − 0.881i)9-s + (−2.08 − 3.12i)10-s + (3.74 − 1.27i)11-s + (0.195 + 1.72i)12-s + (−0.0401 − 0.149i)13-s + (3.76 + 1.27i)14-s + (−6.21 + 1.98i)15-s + (−0.850 + 0.490i)16-s + (1.50 + 3.83i)17-s + ⋯ |
| L(s) = 1 | + (0.699 − 0.0920i)2-s + (0.148 − 0.988i)3-s + (−0.485 + 0.129i)4-s + (−0.745 − 1.51i)5-s + (0.0129 − 0.705i)6-s + (1.35 + 0.666i)7-s + (−0.979 + 0.405i)8-s + (−0.955 − 0.293i)9-s + (−0.660 − 0.988i)10-s + (1.12 − 0.383i)11-s + (0.0564 + 0.499i)12-s + (−0.0111 − 0.0415i)13-s + (1.00 + 0.341i)14-s + (−1.60 + 0.512i)15-s + (−0.212 + 0.122i)16-s + (0.364 + 0.931i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03959 - 0.915413i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03959 - 0.915413i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.257 + 1.71i)T \) |
| 17 | \( 1 + (-1.50 - 3.83i)T \) |
| good | 2 | \( 1 + (-0.989 + 0.130i)T + (1.93 - 0.517i)T^{2} \) |
| 5 | \( 1 + (1.66 + 3.37i)T + (-3.04 + 3.96i)T^{2} \) |
| 7 | \( 1 + (-3.57 - 1.76i)T + (4.26 + 5.55i)T^{2} \) |
| 11 | \( 1 + (-3.74 + 1.27i)T + (8.72 - 6.69i)T^{2} \) |
| 13 | \( 1 + (0.0401 + 0.149i)T + (-11.2 + 6.5i)T^{2} \) |
| 19 | \( 1 + (-3.79 - 1.57i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (0.645 - 0.735i)T + (-3.00 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.325 + 4.96i)T + (-28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (1.57 - 4.65i)T + (-24.5 - 18.8i)T^{2} \) |
| 37 | \( 1 + (0.326 + 1.64i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (4.85 - 0.318i)T + (40.6 - 5.35i)T^{2} \) |
| 43 | \( 1 + (3.14 + 2.41i)T + (11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (0.797 - 2.97i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.152 + 0.367i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (0.226 - 1.71i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (3.50 - 7.09i)T + (-37.1 - 48.3i)T^{2} \) |
| 67 | \( 1 + (9.67 + 5.58i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.55 + 0.707i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-12.4 - 8.29i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (3.82 + 11.2i)T + (-62.6 + 48.0i)T^{2} \) |
| 83 | \( 1 + (-0.0575 - 0.436i)T + (-80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (3.79 - 3.79i)T - 89iT^{2} \) |
| 97 | \( 1 + (-7.24 - 0.474i)T + (96.1 + 12.6i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.59744054568130496116018939660, −11.96586568973221652832612737364, −11.58567670024902900212383421444, −9.103860477702877485195167634737, −8.486402753628034327970380578480, −7.84819825610320788692879939841, −5.86074819472089568838008612076, −4.96058667639346794878461072004, −3.73503319875699623700374477252, −1.40063556975406531420682554436,
3.25622952370343792941379739309, 4.16661005729210281565044366676, 5.12184935788156455573248798474, 6.77316528832180708391570077294, 7.901773445956026364914264395557, 9.274738156170713379574690284806, 10.29903095521607975734153692561, 11.35003013128641371969395246478, 11.81847894346096674367965446025, 13.84497081081219018294050397941
