L(s) = 1 | + (−1 − 1.73i)5-s + (−2 + 3.46i)11-s + (1 + 1.73i)13-s − 2·17-s + 4·19-s + (4 + 6.92i)23-s + (0.500 − 0.866i)25-s + (3 − 5.19i)29-s + (4 + 6.92i)31-s + 6·37-s + (−3 − 5.19i)41-s + (2 − 3.46i)43-s + (3.5 + 6.06i)49-s + 2·53-s + 7.99·55-s + ⋯ |
L(s) = 1 | + (−0.447 − 0.774i)5-s + (−0.603 + 1.04i)11-s + (0.277 + 0.480i)13-s − 0.485·17-s + 0.917·19-s + (0.834 + 1.44i)23-s + (0.100 − 0.173i)25-s + (0.557 − 0.964i)29-s + (0.718 + 1.24i)31-s + 0.986·37-s + (−0.468 − 0.811i)41-s + (0.304 − 0.528i)43-s + (0.5 + 0.866i)49-s + 0.274·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.437677098\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.437677098\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2 + 3.46i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.568509157612770623365410196542, −8.962713037240924111840986637938, −8.029743302419695908881674794434, −7.37570575810996941488469323217, −6.48473118501311957369981862663, −5.24263134539222568747718517582, −4.70111081119873433260608956072, −3.71220130776936427451651741337, −2.42408525011077101508870871840, −1.07256321966832021287789734510,
0.76061982034735608741945139283, 2.69782398504300168800135679028, 3.22829099708825179783608637782, 4.44243641987049034307585558448, 5.46895078929398705798988423245, 6.37222554958022987249871146321, 7.14068162813448002061368247246, 8.052258597447369362397200679185, 8.624330938881923500627405054767, 9.678155260090871653197900647503