L(s) = 1 | − 4i·2-s + 10.0i·3-s − 16·4-s + (7.97 + 55.3i)5-s + 40.0·6-s + 231. i·7-s + 64i·8-s + 142.·9-s + (221. − 31.9i)10-s + 512.·11-s − 160. i·12-s − 758. i·13-s + 926.·14-s + (−553. + 79.8i)15-s + 256·16-s + 1.48e3i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.641i·3-s − 0.5·4-s + (0.142 + 0.989i)5-s + 0.453·6-s + 1.78i·7-s + 0.353i·8-s + 0.587·9-s + (0.699 − 0.100i)10-s + 1.27·11-s − 0.320i·12-s − 1.24i·13-s + 1.26·14-s + (−0.635 + 0.0915i)15-s + 0.250·16-s + 1.24i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 - 0.989i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.142 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.142455167\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.142455167\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 4iT \) |
| 5 | \( 1 + (-7.97 - 55.3i)T \) |
| 23 | \( 1 - 529iT \) |
good | 3 | \( 1 - 10.0iT - 243T^{2} \) |
| 7 | \( 1 - 231. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 512.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 758. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.48e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.81e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 7.88e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.07e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.25e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 2.08e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.94e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.91e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 515. iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 4.23e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 156.T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.85e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 1.76e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.24e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 1.14e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.04e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 7.93e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.58e4iT - 8.58e9T^{2} \) |
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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
−11.56618031067789051347003928328, −10.62059276937288385785818308138, −9.711910117730167327669530701689, −9.088875183791744582387047961597, −7.81632104662425182177990048659, −6.24135129734640286439318293792, −5.38221888879584605811048081122, −3.83102655173735146793915587147, −2.92353753271859659230488933797, −1.61704703917633114686960323385,
0.71161788543120235152365152901, 1.40974368317314602057960069378, 4.01892889094082909173411924995, 4.58785184135246940887906319941, 6.21975243173851956441189856140, 7.24457386569896824549408581717, 7.61147308150574996432358202591, 9.276702369657915928293840106579, 9.605457455626535132240483026986, 11.22925607263479341550502241890