L(s) = 1 | + (1.5 − 0.866i)3-s + 3·5-s + (1.5 − 2.59i)9-s + 3·11-s + (0.5 + 0.866i)13-s + (4.5 − 2.59i)15-s + (−3 − 5.19i)17-s + (2 − 3.46i)19-s − 3·23-s + 4·25-s − 5.19i·27-s + (−1.5 + 2.59i)29-s + (−2.5 + 4.33i)31-s + (4.5 − 2.59i)33-s + (−1 + 1.73i)37-s + ⋯ |
L(s) = 1 | + (0.866 − 0.499i)3-s + 1.34·5-s + (0.5 − 0.866i)9-s + 0.904·11-s + (0.138 + 0.240i)13-s + (1.16 − 0.670i)15-s + (−0.727 − 1.26i)17-s + (0.458 − 0.794i)19-s − 0.625·23-s + 0.800·25-s − 0.999i·27-s + (−0.278 + 0.482i)29-s + (−0.449 + 0.777i)31-s + (0.783 − 0.452i)33-s + (−0.164 + 0.284i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.155610139\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.155610139\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3T + 5T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.5 - 7.79i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.5 - 9.52i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.119132060948563196077013066792, −8.760153169971804146104741828183, −7.44591564765791901932394967454, −6.86019793862356855528174165572, −6.16754609675195071409399296125, −5.17679836486865466942365181029, −4.10053570936901699772942819289, −2.94680238624405992046525258292, −2.13830282938126186905957234257, −1.17537700419931611653622631589,
1.64474442766836725824304879449, 2.24975969097695172950960929349, 3.58315320934921240019072276239, 4.21197945409123824725531504680, 5.46822607169730739532048724547, 6.07132490515510078279087567771, 6.99016945257227287322964106892, 8.093883675189657535525462497639, 8.690412663272820784310503814405, 9.521639218601219668900771416064