L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)11-s − 13-s + 0.999·15-s + (0.5 − 0.866i)17-s + (−0.499 + 0.866i)25-s − 27-s − 29-s + (−0.499 − 0.866i)33-s + (0.5 − 0.866i)39-s + (−0.5 − 0.866i)47-s + (0.499 + 0.866i)51-s + 0.999·55-s + (0.5 + 0.866i)65-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)11-s − 13-s + 0.999·15-s + (0.5 − 0.866i)17-s + (−0.499 + 0.866i)25-s − 27-s − 29-s + (−0.499 − 0.866i)33-s + (0.5 − 0.866i)39-s + (−0.5 − 0.866i)47-s + (0.499 + 0.866i)51-s + 0.999·55-s + (0.5 + 0.866i)65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.04042384498\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04042384498\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + 2T + T^{2} \) |
| 73 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + 2T + T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−9.208004666237471646548035870310, −8.372765247895775922316789846984, −7.40267394829386834699338797138, −7.24501194664841360452335092537, −5.70177951799821100173811433407, −5.21017622692175336726612236322, −4.60341898679440797292524763379, −4.04345882523985575165827446863, −2.87194927327832905627592696237, −1.69810432532893621843676467960,
0.02366696605190162706817059499, 1.51996097849080123173746990361, 2.65682932067581481402282493049, 3.44833510552664789025519452950, 4.35521299984307247806561806836, 5.54640364137133478784855125364, 6.06115396146323053010651760243, 6.80739465675325071096302125763, 7.53005861239001141794397994772, 7.86519428439525497819822141311