L(s) = 1 | + (0.939 + 0.342i)7-s + (−0.173 + 0.984i)13-s + (0.5 + 0.866i)19-s + (0.173 + 0.984i)25-s + (−1.87 + 0.684i)31-s + (0.5 − 0.866i)37-s + (1.53 − 1.28i)43-s + (0.939 + 0.342i)61-s + (−0.173 + 0.984i)67-s + (0.5 + 0.866i)73-s + (−0.173 − 0.984i)79-s + (−0.5 + 0.866i)91-s + (−0.766 + 0.642i)97-s + (−0.766 − 0.642i)103-s + 2·109-s + ⋯ |
L(s) = 1 | + (0.939 + 0.342i)7-s + (−0.173 + 0.984i)13-s + (0.5 + 0.866i)19-s + (0.173 + 0.984i)25-s + (−1.87 + 0.684i)31-s + (0.5 − 0.866i)37-s + (1.53 − 1.28i)43-s + (0.939 + 0.342i)61-s + (−0.173 + 0.984i)67-s + (0.5 + 0.866i)73-s + (−0.173 − 0.984i)79-s + (−0.5 + 0.866i)91-s + (−0.766 + 0.642i)97-s + (−0.766 − 0.642i)103-s + 2·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.344683150\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.344683150\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 7 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 11 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 13 | \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 29 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 31 | \( 1 + (1.87 - 0.684i)T + (0.766 - 0.642i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (-1.53 + 1.28i)T + (0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 61 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 67 | \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
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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
−9.041619208290546965479768969932, −8.340387474599730566045845305326, −7.42707144738471128819462887991, −7.00747096638481633324984414667, −5.72739024959817673521484926831, −5.34046593507801552431614978655, −4.30285893133009566131719603764, −3.56119742818984898967014802761, −2.25909532123487143564689760442, −1.47939413894260555533063065734,
0.926455049033312825485381457739, 2.21064625364155192216402743476, 3.16830575505932132823356250452, 4.25088937548966989785406084900, 4.95402642734335419867295204837, 5.69806309511382242921483132496, 6.61342556824896057230204606674, 7.63246333698419912557163056714, 7.87115430542624607738769798066, 8.826975927145222540572294353549