L(s) = 1 | − i·5-s + 1.41i·7-s − 1.41·11-s − 1.41·13-s − 2·23-s − 25-s + 1.41·35-s − 1.41·37-s − 1.41i·41-s − 1.00·49-s + 1.41i·55-s + 1.41·59-s + 1.41i·65-s − 2.00i·77-s + 1.41i·89-s + ⋯ |
L(s) = 1 | − i·5-s + 1.41i·7-s − 1.41·11-s − 1.41·13-s − 2·23-s − 25-s + 1.41·35-s − 1.41·37-s − 1.41i·41-s − 1.00·49-s + 1.41i·55-s + 1.41·59-s + 1.41i·65-s − 2.00i·77-s + 1.41i·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.05542896878\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05542896878\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
good | 7 | \( 1 - 1.41iT - T^{2} \) |
| 11 | \( 1 + 1.41T + T^{2} \) |
| 13 | \( 1 + 1.41T + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 2T + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + 1.41T + T^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.41T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.41iT - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.264556273116822676624641864792, −8.477094789331404379556607822218, −8.017663034118660330565311762303, −7.20913154763306741845250480100, −5.98259576386807989393472877714, −5.33042603206199091461793389858, −4.98161625492424237018264718628, −3.83748970608071710770929645148, −2.48241400734359867856245209807, −2.05243266777295900016439380181,
0.03051562360724626767281873836, 1.99485879519982795400567975238, 2.86182270781797951769350910301, 3.78910973679140441092664974012, 4.62217260578478365151197680884, 5.51306654170881316723314959061, 6.48308599571793006008193901780, 7.26939292463884475806779312648, 7.64448852680929957597526430039, 8.308763766298565459172886852940