| L(s) = 1 | − 1.41i·2-s − 2.00·4-s − 2.80·5-s + 4.63·7-s + 2.82i·8-s + 3.96i·10-s + 15.7i·11-s − 0.726·13-s − 6.55i·14-s + 4.00·16-s − 22.6·17-s − 5.82i·19-s + 5.60·20-s + 22.2·22-s − 7.17·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 0.560·5-s + 0.661·7-s + 0.353i·8-s + 0.396i·10-s + 1.43i·11-s − 0.0558·13-s − 0.468i·14-s + 0.250·16-s − 1.33·17-s − 0.306i·19-s + 0.280·20-s + 1.01·22-s − 0.312·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 + 0.537i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2034 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8034434568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8034434568\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 113 | \( 1 + (-112. - 5.38i)T \) |
| good | 5 | \( 1 + 2.80T + 25T^{2} \) |
| 7 | \( 1 - 4.63T + 49T^{2} \) |
| 11 | \( 1 - 15.7iT - 121T^{2} \) |
| 13 | \( 1 + 0.726T + 169T^{2} \) |
| 17 | \( 1 + 22.6T + 289T^{2} \) |
| 19 | \( 1 + 5.82iT - 361T^{2} \) |
| 23 | \( 1 + 7.17T + 529T^{2} \) |
| 29 | \( 1 - 29.3T + 841T^{2} \) |
| 31 | \( 1 - 8.55T + 961T^{2} \) |
| 37 | \( 1 + 19.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 56.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 44.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.03T + 2.20e3T^{2} \) |
| 53 | \( 1 + 74.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 84.2T + 3.48e3T^{2} \) |
| 61 | \( 1 - 34.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 70.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 46.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + 58.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 105. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 68.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 116.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 90.6T + 9.40e3T^{2} \) |
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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
−8.660241460982951903003110804183, −7.979005409493081369660006304167, −7.19462494758190207031581886454, −6.34983622909073759128093422889, −4.96928379599805459642698027113, −4.55896524849097482606993671552, −3.72920513668191848140799494331, −2.43390594501064025929387694233, −1.73115461842906499336437603514, −0.23440959392729308527277281957,
1.00515324621468446525697986582, 2.52921022166146818818884210338, 3.74265636446706429350232189624, 4.42946079823538149523877591655, 5.37396272155354978141465757622, 6.16612858009513963835483622748, 6.91138169015755466544078200592, 7.87850692303695819077776403094, 8.383796373240948086059795017299, 8.900182079104510940020367036004
