L(s) = 1 | + (2.72 − 0.773i)2-s + (6.80 − 4.21i)4-s + 20.8i·5-s + 13.9i·7-s + (15.2 − 16.7i)8-s + (16.1 + 56.7i)10-s + 34.5·11-s − 31.3·13-s + (10.7 + 37.8i)14-s + (28.5 − 57.2i)16-s − 34.4i·17-s − 120. i·19-s + (87.7 + 141. i)20-s + (93.8 − 26.7i)22-s + 137.·23-s + ⋯ |
L(s) = 1 | + (0.961 − 0.273i)2-s + (0.850 − 0.526i)4-s + 1.86i·5-s + 0.750i·7-s + (0.673 − 0.738i)8-s + (0.510 + 1.79i)10-s + 0.945·11-s − 0.668·13-s + (0.205 + 0.722i)14-s + (0.445 − 0.895i)16-s − 0.491i·17-s − 1.45i·19-s + (0.981 + 1.58i)20-s + (0.909 − 0.258i)22-s + 1.24·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.526i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.70750 + 0.770263i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.70750 + 0.770263i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.72 + 0.773i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 20.8iT - 125T^{2} \) |
| 7 | \( 1 - 13.9iT - 343T^{2} \) |
| 11 | \( 1 - 34.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 31.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 34.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 120. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 137.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 93.1iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 111. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 146.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 8.44iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 427. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 318.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 291. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 364.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 289.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 305. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 102.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 442.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 245. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 478.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.41e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−13.49888636816011644603241087877, −12.10615459638430798884517502332, −11.34982912154787092339725455734, −10.52665263586442919075859243535, −9.272728149466355635299710882348, −7.09621761383346818200936900149, −6.65658303165758858182999354030, −5.15760060290614537456062882040, −3.38190806725150980036246672811, −2.40815448797661654392573465023,
1.40338959761437196051778358859, 3.91012189583593365728573490993, 4.79002072264481083695991733866, 6.01534289626603080519119348465, 7.52964128626482945440877209462, 8.582168086721674576702352844996, 9.883130149038244995132864489615, 11.48634514596860123649755214270, 12.40806947656613570717609181852, 13.02896302772665116317571194176