| L(s) = 1 | + 8.89·5-s − 4.08·7-s − 4.37·11-s − 13.7i·13-s − 14.0·17-s + (−11.6 + 15.0i)19-s − 6.99·23-s + 54.0·25-s − 31.8i·29-s + 12.3i·31-s − 36.3·35-s − 4.89i·37-s − 73.1i·41-s + 23.8·43-s − 43.3·47-s + ⋯ |
| L(s) = 1 | + 1.77·5-s − 0.584·7-s − 0.397·11-s − 1.05i·13-s − 0.828·17-s + (−0.611 + 0.791i)19-s − 0.304·23-s + 2.16·25-s − 1.09i·29-s + 0.398i·31-s − 1.03·35-s − 0.132i·37-s − 1.78i·41-s + 0.555·43-s − 0.922·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.328680400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.328680400\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (11.6 - 15.0i)T \) |
| good | 5 | \( 1 - 8.89T + 25T^{2} \) |
| 7 | \( 1 + 4.08T + 49T^{2} \) |
| 11 | \( 1 + 4.37T + 121T^{2} \) |
| 13 | \( 1 + 13.7iT - 169T^{2} \) |
| 17 | \( 1 + 14.0T + 289T^{2} \) |
| 23 | \( 1 + 6.99T + 529T^{2} \) |
| 29 | \( 1 + 31.8iT - 841T^{2} \) |
| 31 | \( 1 - 12.3iT - 961T^{2} \) |
| 37 | \( 1 + 4.89iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 73.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 23.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 43.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 65.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 62.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 70.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 126. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 44.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 0.118T + 5.32e3T^{2} \) |
| 79 | \( 1 - 19.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 97.8T + 6.88e3T^{2} \) |
| 89 | \( 1 + 15.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 6.98iT - 9.40e3T^{2} \) |
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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
−8.490902205030001211812466571543, −7.64382971445575470302344884329, −6.57426221329623231189762404480, −6.05241259089962453461271043376, −5.50323763418555722437374970781, −4.57746861009934996502202962687, −3.33961000042403181082142407233, −2.44355935721116753994865041034, −1.71890663628336659611908172469, −0.26615013656272162272526789851,
1.40105046443239875838477532808, 2.26195187538765121606417266330, 2.97446672396837553275804104828, 4.35662601494213312681614219443, 5.07245792320798402546767685710, 6.01193799012663062412269970007, 6.53613177953476243091407216639, 7.08252123409662430891658702564, 8.422775066053179690707161004722, 9.069622898510337680563650329200
