L(s) = 1 | + (−0.365 − 0.930i)2-s + (−0.733 + 0.680i)4-s + (0.900 + 0.433i)8-s + (0.365 + 0.930i)11-s + (−0.988 − 0.149i)13-s + (0.0747 − 0.997i)16-s + (−0.222 + 0.974i)17-s − 19-s + (0.733 − 0.680i)22-s + (0.733 − 0.680i)23-s + (0.222 + 0.974i)26-s + (−0.733 − 0.680i)29-s + (0.5 + 0.866i)31-s + (−0.955 + 0.294i)32-s + (0.988 − 0.149i)34-s + ⋯ |
L(s) = 1 | + (−0.365 − 0.930i)2-s + (−0.733 + 0.680i)4-s + (0.900 + 0.433i)8-s + (0.365 + 0.930i)11-s + (−0.988 − 0.149i)13-s + (0.0747 − 0.997i)16-s + (−0.222 + 0.974i)17-s − 19-s + (0.733 − 0.680i)22-s + (0.733 − 0.680i)23-s + (0.222 + 0.974i)26-s + (−0.733 − 0.680i)29-s + (0.5 + 0.866i)31-s + (−0.955 + 0.294i)32-s + (0.988 − 0.149i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.04043506482 - 0.3092626142i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04043506482 - 0.3092626142i\) |
\(L(1)\) |
\(\approx\) |
\(0.6784897245 - 0.1931711126i\) |
\(L(1)\) |
\(\approx\) |
\(0.6784897245 - 0.1931711126i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.365 - 0.930i)T \) |
| 11 | \( 1 + (0.365 + 0.930i)T \) |
| 13 | \( 1 + (-0.988 - 0.149i)T \) |
| 17 | \( 1 + (-0.222 + 0.974i)T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (0.733 - 0.680i)T \) |
| 29 | \( 1 + (-0.733 - 0.680i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.222 - 0.974i)T \) |
| 41 | \( 1 + (-0.0747 - 0.997i)T \) |
| 43 | \( 1 + (-0.0747 + 0.997i)T \) |
| 47 | \( 1 + (0.365 + 0.930i)T \) |
| 53 | \( 1 + (0.222 + 0.974i)T \) |
| 59 | \( 1 + (-0.826 - 0.563i)T \) |
| 61 | \( 1 + (0.733 + 0.680i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.222 - 0.974i)T \) |
| 73 | \( 1 + (0.623 + 0.781i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.988 + 0.149i)T \) |
| 89 | \( 1 + (-0.623 - 0.781i)T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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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
−19.6432835968995329227531169385, −18.98247381551513690796562968399, −18.486571919836290713775344574785, −17.54633916712485195249547886462, −16.7850579692982414457298119129, −16.59765378562880445501186781713, −15.44197243376669731399183555918, −15.01019109622544668162857952363, −14.188423231233686354623822410460, −13.5356230755788572094244119660, −12.85317565737940494177674673038, −11.70710041226553941333264593239, −11.05813690696676118423128935041, −10.07143382069607738523211774909, −9.40360803472055034836575916644, −8.729880082272696403385756730898, −7.982428378179463856841519808, −7.0975932221186601549565825046, −6.57820438841846701783989794174, −5.61031300187796075723635097742, −4.9518403782826657913293379777, −4.10316045407302186000523435000, −3.03777895405420833300346012428, −1.87174635719174544115577750924, −0.72759912181057223684526225008,
0.08338609827952914296293025576, 1.26942975244398041289647099276, 2.166981813669934442053843511514, 2.77712298261273716857790553776, 4.05392000598701880727123984780, 4.42244926134364619746701765327, 5.43216656851586752192459514181, 6.62232144884154330101449127273, 7.40656272275323386144607306441, 8.22522449403316412667809213911, 9.02260556683095351141602852761, 9.68364221950108362735921854088, 10.466044788131571970007808948291, 10.99361093164122119609834227214, 11.99664426327407298881228023268, 12.646386834679085804055819985173, 12.95828675029567418235841193433, 14.17779884240352036354155024883, 14.74426940474172798933720079855, 15.5333865044841912765331355336, 16.7440636314344957090863387018, 17.267045443655206293413477252213, 17.67948232462473082127406801830, 18.70256892397553020808071857242, 19.35016452380000066978876164732