L(s) = 1 | + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)3-s + (0.173 − 0.984i)4-s + (0.5 + 0.866i)6-s + (0.500 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (1.70 + 0.984i)11-s + (−0.939 − 0.342i)12-s + (−0.939 − 0.342i)16-s + (1.70 − 0.300i)17-s + (0.939 − 0.342i)18-s + (−0.766 − 0.642i)19-s + (−1.93 + 0.342i)22-s + (0.939 − 0.342i)24-s + (−0.173 + 0.984i)25-s + ⋯ |
L(s) = 1 | + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)3-s + (0.173 − 0.984i)4-s + (0.5 + 0.866i)6-s + (0.500 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (1.70 + 0.984i)11-s + (−0.939 − 0.342i)12-s + (−0.939 − 0.342i)16-s + (1.70 − 0.300i)17-s + (0.939 − 0.342i)18-s + (−0.766 − 0.642i)19-s + (−1.93 + 0.342i)22-s + (0.939 − 0.342i)24-s + (−0.173 + 0.984i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8600033052\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8600033052\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 3 | \( 1 + (-0.173 + 0.984i)T \) |
| 19 | \( 1 + (0.766 + 0.642i)T \) |
good | 5 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (-1.70 - 0.984i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 17 | \( 1 + (-1.70 + 0.300i)T + (0.939 - 0.342i)T^{2} \) |
| 23 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (0.347 + 1.96i)T + (-0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 59 | \( 1 + (1.76 + 0.642i)T + (0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 67 | \( 1 + (-1.26 + 1.50i)T + (-0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 73 | \( 1 + (0.266 - 0.223i)T + (0.173 - 0.984i)T^{2} \) |
| 79 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 83 | \( 1 + 0.684iT - T^{2} \) |
| 89 | \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 97 | \( 1 + (0.826 + 0.984i)T + (-0.173 + 0.984i)T^{2} \) |
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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
−9.360139000631850578730220303796, −9.008420463717439447891367712529, −7.983982940982851662782520973348, −7.27884769090720985133942012986, −6.74127486401660427026520170328, −5.95649995292668652955261855768, −4.98392632721888371726582038795, −3.60390856250687788681105970493, −2.08752825016523120377234869709, −1.16968622321874711850254214133,
1.31609563301681678471523518088, 2.85391725971619545374404297672, 3.71839673722011985594329258933, 4.27704657879429193890976536399, 5.75842957773803593876708568689, 6.51178432422466665905494807424, 7.87693196738849798049886877349, 8.424970382151501638330766185262, 9.166119299103288028376589791361, 9.815181878303247341839222315888