L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)9-s + (−1.5 − 0.866i)11-s + 17-s − 1.73i·19-s + (0.5 − 0.866i)25-s + 0.999·27-s + (1.5 − 0.866i)33-s + (−0.5 − 0.866i)41-s + (1.5 + 0.866i)43-s + (−0.5 − 0.866i)49-s + (−0.5 + 0.866i)51-s + (1.49 + 0.866i)57-s + (−1.5 + 0.866i)59-s + (1.5 − 0.866i)67-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)9-s + (−1.5 − 0.866i)11-s + 17-s − 1.73i·19-s + (0.5 − 0.866i)25-s + 0.999·27-s + (1.5 − 0.866i)33-s + (−0.5 − 0.866i)41-s + (1.5 + 0.866i)43-s + (−0.5 − 0.866i)49-s + (−0.5 + 0.866i)51-s + (1.49 + 0.866i)57-s + (−1.5 + 0.866i)59-s + (1.5 − 0.866i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7685759548\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7685759548\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 + 1.73iT - T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - 2T + T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.163812027834394774548878849740, −8.440198218013408817418866236911, −7.65484550460326826376738680975, −6.65135054077798821651233858875, −5.76704191023427785652996088699, −5.15989707299736134545521197121, −4.45583216891991361716697981538, −3.27240944795662776286839555987, −2.63919403612049682151273130400, −0.58801804584055334542907052762,
1.36416374581958776184885824734, 2.36554791937411813960380152206, 3.40602800127778600023832834874, 4.74848220071396319869391504358, 5.45497924284220093302993069538, 6.06769170648307973716908360533, 7.10957752643818980496318820883, 7.79211370248316861524361187863, 8.076358446923671498948303426255, 9.315443038155154327358345653219