L(s) = 1 | − 1.41i·2-s + (2.93 − 0.628i)3-s − 2.00·4-s − 3.99i·5-s + (−0.888 − 4.14i)6-s + 0.373·7-s + 2.82i·8-s + (8.21 − 3.68i)9-s − 5.64·10-s + 4.79i·11-s + (−5.86 + 1.25i)12-s − 3.60·13-s − 0.527i·14-s + (−2.50 − 11.7i)15-s + 4.00·16-s + 6.00i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.977 − 0.209i)3-s − 0.500·4-s − 0.798i·5-s + (−0.148 − 0.691i)6-s + 0.0533·7-s + 0.353i·8-s + (0.912 − 0.409i)9-s − 0.564·10-s + 0.435i·11-s + (−0.488 + 0.104i)12-s − 0.277·13-s − 0.0377i·14-s + (−0.167 − 0.780i)15-s + 0.250·16-s + 0.353i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.23101 - 0.995356i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23101 - 0.995356i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (-2.93 + 0.628i)T \) |
| 13 | \( 1 + 3.60T \) |
good | 5 | \( 1 + 3.99iT - 25T^{2} \) |
| 7 | \( 1 - 0.373T + 49T^{2} \) |
| 11 | \( 1 - 4.79iT - 121T^{2} \) |
| 17 | \( 1 - 6.00iT - 289T^{2} \) |
| 19 | \( 1 + 3.50T + 361T^{2} \) |
| 23 | \( 1 - 34.1iT - 529T^{2} \) |
| 29 | \( 1 - 48.5iT - 841T^{2} \) |
| 31 | \( 1 - 1.30T + 961T^{2} \) |
| 37 | \( 1 + 60.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 49.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 47.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 54.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 88.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 61.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 106.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 72.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 104. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 52.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 11.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 75.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 54.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 21.0T + 9.40e3T^{2} \) |
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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
−13.77932811391930480127693844711, −12.81581862491723252032654329916, −12.12061100345287943604015901243, −10.51611557747142517560261908973, −9.334531372059717840773327962651, −8.561956535742104465468906783396, −7.23485982831495060099439589423, −5.03823852341668241019699419058, −3.54429702755999222324054635915, −1.70162412830020832503454114860,
2.85243793873880639229034599122, 4.48374017510074106412204355826, 6.37397911651419801505344491103, 7.55075675828092742593234698426, 8.596365379939922005069202467745, 9.779294897542435413790745663753, 10.87472506217850142632621207235, 12.59175524909957484406686837797, 13.83871824839259320775701046433, 14.44312310094873831673835316150