L(s) = 1 | + (1.40 + 0.174i)2-s − i·3-s + (1.93 + 0.489i)4-s + 2.63i·5-s + (0.174 − 1.40i)6-s − 0.526·7-s + (2.63 + 1.02i)8-s − 9-s + (−0.459 + 3.69i)10-s + 4.16·11-s + (0.489 − 1.93i)12-s − 1.45·13-s + (−0.738 − 0.0918i)14-s + 2.63·15-s + (3.52 + 1.89i)16-s − 7.00i·17-s + ⋯ |
L(s) = 1 | + (0.992 + 0.123i)2-s − 0.577i·3-s + (0.969 + 0.244i)4-s + 1.17i·5-s + (0.0712 − 0.572i)6-s − 0.198·7-s + (0.931 + 0.362i)8-s − 0.333·9-s + (−0.145 + 1.16i)10-s + 1.25·11-s + (0.141 − 0.559i)12-s − 0.403·13-s + (−0.197 − 0.0245i)14-s + 0.680·15-s + (0.880 + 0.474i)16-s − 1.69i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.24676 + 0.270099i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.24676 + 0.270099i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.40 - 0.174i)T \) |
| 3 | \( 1 + iT \) |
| 23 | \( 1 + (2.24 + 4.23i)T \) |
good | 5 | \( 1 - 2.63iT - 5T^{2} \) |
| 7 | \( 1 + 0.526T + 7T^{2} \) |
| 11 | \( 1 - 4.16T + 11T^{2} \) |
| 13 | \( 1 + 1.45T + 13T^{2} \) |
| 17 | \( 1 + 7.00iT - 17T^{2} \) |
| 19 | \( 1 + 7.45T + 19T^{2} \) |
| 29 | \( 1 + 3.70T + 29T^{2} \) |
| 31 | \( 1 - 8.01iT - 31T^{2} \) |
| 37 | \( 1 + 2.39iT - 37T^{2} \) |
| 41 | \( 1 + 4.81T + 41T^{2} \) |
| 43 | \( 1 - 8.10T + 43T^{2} \) |
| 47 | \( 1 - 1.90iT - 47T^{2} \) |
| 53 | \( 1 - 6.35iT - 53T^{2} \) |
| 59 | \( 1 + 7.78iT - 59T^{2} \) |
| 61 | \( 1 - 0.558iT - 61T^{2} \) |
| 67 | \( 1 + 7.34T + 67T^{2} \) |
| 71 | \( 1 + 8.61iT - 71T^{2} \) |
| 73 | \( 1 - 0.375T + 73T^{2} \) |
| 79 | \( 1 + 0.238T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 + 9.60iT - 89T^{2} \) |
| 97 | \( 1 + 3.43iT - 97T^{2} \) |
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
−12.05475131447386602481536175841, −11.21309882575696986553872929946, −10.40904921075305133822422559392, −8.985668367778562033509109374574, −7.52981048026274730771853821131, −6.73988654633824311685786388386, −6.24052411162765138237876292622, −4.66429168555690157193011590198, −3.32915077602312490955829996277, −2.22993497343052322482260095142,
1.81295907926523132908368528950, 3.84183145315258650661656186108, 4.36484532986788351931539996631, 5.65270872639253509668139286302, 6.47785670566429107015178697574, 8.051239373589724217904154618908, 9.068379065653257541652909124548, 10.05757621420644482154250835931, 11.10643906020078352596431907671, 12.04200991067185071777110441465