L(s) = 1 | + (−0.104 + 0.994i)2-s + (−0.978 − 0.207i)4-s + (0.104 + 0.994i)5-s + (−0.669 + 0.743i)7-s + (0.309 − 0.951i)8-s − 10-s + (−0.913 + 0.406i)13-s + (−0.669 − 0.743i)14-s + (0.913 + 0.406i)16-s + (−0.809 + 0.587i)17-s + (−0.309 + 0.951i)19-s + (0.104 − 0.994i)20-s + (0.5 − 0.866i)23-s + (−0.978 + 0.207i)25-s + (−0.309 − 0.951i)26-s + ⋯ |
L(s) = 1 | + (−0.104 + 0.994i)2-s + (−0.978 − 0.207i)4-s + (0.104 + 0.994i)5-s + (−0.669 + 0.743i)7-s + (0.309 − 0.951i)8-s − 10-s + (−0.913 + 0.406i)13-s + (−0.669 − 0.743i)14-s + (0.913 + 0.406i)16-s + (−0.809 + 0.587i)17-s + (−0.309 + 0.951i)19-s + (0.104 − 0.994i)20-s + (0.5 − 0.866i)23-s + (−0.978 + 0.207i)25-s + (−0.309 − 0.951i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1027327911 + 0.6769040741i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1027327911 + 0.6769040741i\) |
\(L(1)\) |
\(\approx\) |
\(0.5355217282 + 0.5733696634i\) |
\(L(1)\) |
\(\approx\) |
\(0.5355217282 + 0.5733696634i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.104 + 0.994i)T \) |
| 5 | \( 1 + (0.104 + 0.994i)T \) |
| 7 | \( 1 + (-0.669 + 0.743i)T \) |
| 13 | \( 1 + (-0.913 + 0.406i)T \) |
| 17 | \( 1 + (-0.809 + 0.587i)T \) |
| 19 | \( 1 + (-0.309 + 0.951i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.669 - 0.743i)T \) |
| 31 | \( 1 + (0.913 - 0.406i)T \) |
| 37 | \( 1 + (0.309 + 0.951i)T \) |
| 41 | \( 1 + (0.669 + 0.743i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.978 - 0.207i)T \) |
| 53 | \( 1 + (0.809 + 0.587i)T \) |
| 59 | \( 1 + (0.978 + 0.207i)T \) |
| 61 | \( 1 + (-0.913 - 0.406i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.809 - 0.587i)T \) |
| 73 | \( 1 + (-0.309 - 0.951i)T \) |
| 79 | \( 1 + (0.104 - 0.994i)T \) |
| 83 | \( 1 + (0.913 + 0.406i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (-0.104 + 0.994i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−29.28901000697245068676752131670, −28.79341178358813149385972628352, −27.56740883832876472235057278804, −26.75955944474319560852697185068, −25.524889923716495272318165192454, −24.16562949362208223488144451077, −23.09742333522216095614927508544, −22.03703000603801219520174799989, −20.973062008878554788576938054436, −19.860033282008740590748588453094, −19.50794430768432456828633257688, −17.7323492083498155869267563230, −17.06127049654748668739098303701, −15.73364444329952672509868261234, −13.9146409841631858324621277834, −13.06855715467739282096163408102, −12.19559521047561540123830589484, −10.83061002634953723429738366506, −9.66522108114257122094772151100, −8.80514116963747808979347241388, −7.25402281985945165578674340329, −5.21786736602733331456763481974, −4.15195702566572105970375016469, −2.58543558663710666161186632005, −0.7356281352931826759879323811,
2.61693545948309226030581450456, 4.33171969394337556628597711032, 6.0187149684587459981610691531, 6.70527854860920907109734931978, 8.07895933807771964512171448348, 9.4083079206820185381408015005, 10.39976964429902517995820463475, 12.14738795960630692470354995833, 13.431582564642454863093158772460, 14.67961772941120919130750149750, 15.30125229456478223020397282293, 16.55609302268999328928452017038, 17.66294809341358554915654561622, 18.77075045519516981941551537137, 19.37601875120588994178669261276, 21.496177106512396877335063560254, 22.35008447465764263518990243882, 23.08187053774111477615039858864, 24.47381748944896630806548913082, 25.283343142210717317950341823515, 26.3324030590565563459093777785, 26.92870706734892090913698992651, 28.30185830775350070831282636439, 29.319008678701464933376351393141, 30.77181227824400208197677695101