| L(s) = 1 | − i·5-s + 2.44i·7-s − 5.91·11-s + 6.24·13-s − 4.89i·19-s − 1.43·23-s − 25-s − 6i·29-s + 1.43i·31-s + 2.44·35-s + 2.24·37-s + 4.24i·41-s − 11.8i·43-s + 11.8·47-s + 1.00·49-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s + 0.925i·7-s − 1.78·11-s + 1.73·13-s − 1.12i·19-s − 0.299·23-s − 0.200·25-s − 1.11i·29-s + 0.257i·31-s + 0.414·35-s + 0.368·37-s + 0.662i·41-s − 1.80i·43-s + 1.72·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.600991395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.600991395\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.44iT - 7T^{2} \) |
| 11 | \( 1 + 5.91T + 11T^{2} \) |
| 13 | \( 1 - 6.24T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 4.89iT - 19T^{2} \) |
| 23 | \( 1 + 1.43T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 1.43iT - 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 - 4.24iT - 41T^{2} \) |
| 43 | \( 1 + 11.8iT - 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 8.48iT - 53T^{2} \) |
| 59 | \( 1 + 5.91T + 59T^{2} \) |
| 61 | \( 1 - 6.48T + 61T^{2} \) |
| 67 | \( 1 - 6.92iT - 67T^{2} \) |
| 71 | \( 1 - 2.86T + 71T^{2} \) |
| 73 | \( 1 - 6.48T + 73T^{2} \) |
| 79 | \( 1 + 8.36iT - 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 7.75iT - 89T^{2} \) |
| 97 | \( 1 - 6.48T + 97T^{2} \) |
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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
−8.719011986274831246065131323302, −8.082267755414007660276754443574, −7.33561710794759757300073891620, −6.15896092195634071360668931559, −5.66840528763991646318644439342, −4.93914284768615839047864370492, −3.97393941409191034726514075222, −2.85582097531899810086706984621, −2.12757681622768363121129327376, −0.63889746964627567283943226393,
0.951956642970081034380029446660, 2.21721675783344314544122012747, 3.35836803534348389688982963610, 3.90911990765912419817557869366, 5.02203941852048258357428198032, 5.85558807561979225828373930065, 6.52191734801970102797604121775, 7.53977268079183320961904882422, 7.938403335725539021081424585772, 8.689128719369735609967315046999
