L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s + 0.999·8-s + (−0.499 − 0.866i)9-s − 0.999·12-s + 13-s + (−0.5 + 0.866i)16-s + 0.999·18-s − 23-s + (0.5 − 0.866i)24-s − 25-s + (−0.5 + 0.866i)26-s − 0.999·27-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s + 0.999·8-s + (−0.499 − 0.866i)9-s − 0.999·12-s + 13-s + (−0.5 + 0.866i)16-s + 0.999·18-s − 23-s + (0.5 − 0.866i)24-s − 25-s + (−0.5 + 0.866i)26-s − 0.999·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6692191272\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6692191272\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 29 | \( 1 - 1.73iT - T^{2} \) |
| 31 | \( 1 - 1.73iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.73iT - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - 2T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 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
−12.28369280332853424723524056946, −11.07051760291761965682653486424, −9.982084605689207700190134519129, −8.833721293587256502114122514770, −8.308550141957662946975868327424, −7.23666090356560335081352254691, −6.43976013446279871700156380824, −5.40028989208129575099032959000, −3.65993122708462194599119316364, −1.64425333060033344749402799769,
2.21543474792341932808993074555, 3.60892462546021914698239996602, 4.41958513492956477317319727359, 5.99130813376410036201674637656, 7.88036979286137812645767799563, 8.358998276420851463323738329659, 9.655080417764662182485215472039, 9.938337436989471838831515276625, 11.23605330536679643242520830880, 11.63075216399526877768610488667