L(s) = 1 | − 1.22i·2-s + (0.136 − 1.72i)3-s + 0.490·4-s − 3.00·5-s + (−2.12 − 0.167i)6-s + (0.249 − 2.63i)7-s − 3.05i·8-s + (−2.96 − 0.470i)9-s + 3.69i·10-s + 2.71i·11-s + (0.0668 − 0.846i)12-s + 2.99i·13-s + (−3.23 − 0.306i)14-s + (−0.409 + 5.18i)15-s − 2.77·16-s + 17-s + ⋯ |
L(s) = 1 | − 0.868i·2-s + (0.0787 − 0.996i)3-s + 0.245·4-s − 1.34·5-s + (−0.866 − 0.0684i)6-s + (0.0944 − 0.995i)7-s − 1.08i·8-s + (−0.987 − 0.156i)9-s + 1.16i·10-s + 0.817i·11-s + (0.0193 − 0.244i)12-s + 0.830i·13-s + (−0.864 − 0.0820i)14-s + (−0.105 + 1.33i)15-s − 0.694·16-s + 0.242·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0903066 + 1.03919i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0903066 + 1.03919i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.136 + 1.72i)T \) |
| 7 | \( 1 + (-0.249 + 2.63i)T \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 + 1.22iT - 2T^{2} \) |
| 5 | \( 1 + 3.00T + 5T^{2} \) |
| 11 | \( 1 - 2.71iT - 11T^{2} \) |
| 13 | \( 1 - 2.99iT - 13T^{2} \) |
| 19 | \( 1 + 6.46iT - 19T^{2} \) |
| 23 | \( 1 + 0.531iT - 23T^{2} \) |
| 29 | \( 1 + 8.12iT - 29T^{2} \) |
| 31 | \( 1 - 4.76iT - 31T^{2} \) |
| 37 | \( 1 - 0.597T + 37T^{2} \) |
| 41 | \( 1 + 4.72T + 41T^{2} \) |
| 43 | \( 1 - 8.79T + 43T^{2} \) |
| 47 | \( 1 - 5.20T + 47T^{2} \) |
| 53 | \( 1 + 1.86iT - 53T^{2} \) |
| 59 | \( 1 - 14.9T + 59T^{2} \) |
| 61 | \( 1 + 8.34iT - 61T^{2} \) |
| 67 | \( 1 - 6.91T + 67T^{2} \) |
| 71 | \( 1 + 13.5iT - 71T^{2} \) |
| 73 | \( 1 + 4.26iT - 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 3.09T + 83T^{2} \) |
| 89 | \( 1 - 9.36T + 89T^{2} \) |
| 97 | \( 1 - 5.63iT - 97T^{2} \) |
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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
−11.39056463091312888708480935297, −10.37977534675787046522221050409, −9.166584022815000983132878427361, −7.87617425067045282977335479998, −7.19507194314205932519304663991, −6.62624427194283328400909567727, −4.54803525005967410557135476913, −3.58473671400073277282828090309, −2.24106003454082151270533782981, −0.70191263744714947815682674104,
2.86864972873841162550747490133, 3.90909210861899381004210830069, 5.37795286255109886932526700893, 5.87808994218932171058678176064, 7.41717084005351193550555319884, 8.305974302594526315340349342039, 8.681219398511096554265826410392, 10.16249612960593301610282135526, 11.16017573310516414660547517693, 11.68939492226865921915071140951