L(s) = 1 | + (−2 + 2.23i)3-s + 8·7-s + (−1.00 − 8.94i)9-s − 8.94i·11-s + 12·13-s + 31.3i·17-s + 6·19-s + (−16 + 17.8i)21-s + 4.47i·23-s + (22.0 + 15.6i)27-s + 26.8i·29-s + 34·31-s + (20.0 + 17.8i)33-s + 44·37-s + (−24 + 26.8i)39-s + ⋯ |
L(s) = 1 | + (−0.666 + 0.745i)3-s + 1.14·7-s + (−0.111 − 0.993i)9-s − 0.813i·11-s + 0.923·13-s + 1.84i·17-s + 0.315·19-s + (−0.761 + 0.851i)21-s + 0.194i·23-s + (0.814 + 0.579i)27-s + 0.925i·29-s + 1.09·31-s + (0.606 + 0.542i)33-s + 1.18·37-s + (−0.615 + 0.688i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.36967 + 0.612538i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36967 + 0.612538i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2 - 2.23i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 8T + 49T^{2} \) |
| 11 | \( 1 + 8.94iT - 121T^{2} \) |
| 13 | \( 1 - 12T + 169T^{2} \) |
| 17 | \( 1 - 31.3iT - 289T^{2} \) |
| 19 | \( 1 - 6T + 361T^{2} \) |
| 23 | \( 1 - 4.47iT - 529T^{2} \) |
| 29 | \( 1 - 26.8iT - 841T^{2} \) |
| 31 | \( 1 - 34T + 961T^{2} \) |
| 37 | \( 1 - 44T + 1.36e3T^{2} \) |
| 41 | \( 1 - 17.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 28T + 1.84e3T^{2} \) |
| 47 | \( 1 + 4.47iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 40.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 98.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 74T + 3.72e3T^{2} \) |
| 67 | \( 1 + 92T + 4.48e3T^{2} \) |
| 71 | \( 1 - 53.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 56T + 5.32e3T^{2} \) |
| 79 | \( 1 - 78T + 6.24e3T^{2} \) |
| 83 | \( 1 + 102. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 17.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 32T + 9.40e3T^{2} \) |
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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
−11.26994784482319703813811730186, −10.96985493007450259523492845398, −9.980878689875229571322506235101, −8.693072291846072503169515899157, −8.095618040348254810084468408010, −6.42959859562964145080900312788, −5.63854645933505780650899994485, −4.52428353551788446656738845372, −3.47457750421027810608906245331, −1.28691995407705465160167878900,
1.00875094651662017529541759925, 2.41276010146875839090153387652, 4.49265375350118589065458508438, 5.32712756576990242069786800323, 6.54725523496534546920572089459, 7.51682248996287130635354395387, 8.259450971420396431104118184756, 9.570171244330608412569500349874, 10.72481616995928590502940866135, 11.63127316357116860112761318111