| L(s) = 1 | + (1.67 + 1.09i)2-s + (1.60 + 3.66i)4-s − 5.73i·5-s − 2.64i·7-s + (−1.30 + 7.89i)8-s + (6.26 − 9.60i)10-s + 1.40·11-s − 19.0i·13-s + (2.89 − 4.43i)14-s + (−10.8 + 11.7i)16-s + 32.2·17-s + 12.5·19-s + (20.9 − 9.22i)20-s + (2.34 + 1.53i)22-s − 15.8i·23-s + ⋯ |
| L(s) = 1 | + (0.837 + 0.546i)2-s + (0.402 + 0.915i)4-s − 1.14i·5-s − 0.377i·7-s + (−0.163 + 0.986i)8-s + (0.626 − 0.960i)10-s + 0.127·11-s − 1.46i·13-s + (0.206 − 0.316i)14-s + (−0.676 + 0.736i)16-s + 1.89·17-s + 0.661·19-s + (1.04 − 0.461i)20-s + (0.106 + 0.0696i)22-s − 0.690i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.941504485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.941504485\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.67 - 1.09i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 5 | \( 1 + 5.73iT - 25T^{2} \) |
| 11 | \( 1 - 1.40T + 121T^{2} \) |
| 13 | \( 1 + 19.0iT - 169T^{2} \) |
| 17 | \( 1 - 32.2T + 289T^{2} \) |
| 19 | \( 1 - 12.5T + 361T^{2} \) |
| 23 | \( 1 + 15.8iT - 529T^{2} \) |
| 29 | \( 1 - 3.29iT - 841T^{2} \) |
| 31 | \( 1 - 22.6iT - 961T^{2} \) |
| 37 | \( 1 + 54.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 7.59T + 1.68e3T^{2} \) |
| 43 | \( 1 + 20.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 21.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 0.356iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 26.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 86.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 114.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 104. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 24.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 117. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 79.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 2.66T + 7.92e3T^{2} \) |
| 97 | \( 1 + 52.0T + 9.40e3T^{2} \) |
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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
−10.76255894275293133491550413671, −9.785669000376528958179834839798, −8.560364576435889937479436668406, −7.934324188688711808514069555327, −7.06300348549535568381366028692, −5.58488333297618418197620109474, −5.28479919568183627128027822252, −4.02775931583445146509666251141, −2.99730389107565193963972917813, −1.00827780681119513909422890880,
1.57743114069932614694482936798, 2.89819682767911538875559219833, 3.68844462087773615435812471798, 4.99967949760488117609311303234, 6.05342206528061540211027066740, 6.81702541826411404525519862646, 7.78805894459552996312223513099, 9.422836733175135654157318317656, 9.926026718859203271402303002664, 10.96064935879105424438718730134
