| L(s) = 1 | + 1.41·2-s + 2.00·4-s + 0.242i·7-s + 2.82·8-s + 3i·11-s + 3.48i·13-s + 0.343i·14-s + 4.00·16-s + 7.75·17-s + 8.97·19-s + 4.24i·22-s + 6.51·23-s + 4.92i·26-s + 0.485i·28-s − 7.02i·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s + 0.0346i·7-s + 0.353·8-s + 0.272i·11-s + 0.268i·13-s + 0.0245i·14-s + 0.250·16-s + 0.456·17-s + 0.472·19-s + 0.192i·22-s + 0.283·23-s + 0.189i·26-s + 0.0173i·28-s − 0.242i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.257814184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.257814184\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 0.242iT - 49T^{2} \) |
| 11 | \( 1 - 3iT - 121T^{2} \) |
| 13 | \( 1 - 3.48iT - 169T^{2} \) |
| 17 | \( 1 - 7.75T + 289T^{2} \) |
| 19 | \( 1 - 8.97T + 361T^{2} \) |
| 23 | \( 1 - 6.51T + 529T^{2} \) |
| 29 | \( 1 + 7.02iT - 841T^{2} \) |
| 31 | \( 1 - 29.2T + 961T^{2} \) |
| 37 | \( 1 - 36.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 57.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 11.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 6.51T + 2.20e3T^{2} \) |
| 53 | \( 1 - 26.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 84.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 57.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 97.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 72.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 106. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 78.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 25.8T + 6.88e3T^{2} \) |
| 89 | \( 1 + 31.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 110. iT - 9.40e3T^{2} \) |
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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
−9.668061223241981569907417089065, −8.567133480960075420611346937085, −7.75571086389005315165509335033, −6.89458964042646176499733975432, −6.13320925402424089444108871949, −5.18644062820703541899226561918, −4.43029795949123644842450089090, −3.40424794667431239536045176719, −2.44937787369651432927177471221, −1.13703170693518623283675569779,
0.861038447274663566414108221880, 2.29476420708451471104229228703, 3.31263159847119545439473121373, 4.16734789223849311058451103686, 5.25461914403155419565409013081, 5.83162719705301201620873710518, 6.88005469408458855949013159467, 7.57161253191095844840239032605, 8.497300260886370149295228171437, 9.372302528051965484251371620452
