L(s) = 1 | − 3.16·5-s + (2 − 1.73i)7-s − 6.32·11-s + (−1.58 + 2.73i)17-s + (3.5 + 6.06i)19-s + 3.16·23-s + 5.00·25-s + (1.58 + 2.73i)29-s + (−1.5 − 2.59i)31-s + (−6.32 + 5.47i)35-s + (2 + 3.46i)37-s + (4.74 − 8.21i)41-s + (−2.5 − 4.33i)43-s + (4.74 − 8.21i)47-s + (1.00 − 6.92i)49-s + ⋯ |
L(s) = 1 | − 1.41·5-s + (0.755 − 0.654i)7-s − 1.90·11-s + (−0.383 + 0.664i)17-s + (0.802 + 1.39i)19-s + 0.659·23-s + 1.00·25-s + (0.293 + 0.508i)29-s + (−0.269 − 0.466i)31-s + (−1.06 + 0.925i)35-s + (0.328 + 0.569i)37-s + (0.740 − 1.28i)41-s + (−0.381 − 0.660i)43-s + (0.691 − 1.19i)47-s + (0.142 − 0.989i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.099789717\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.099789717\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 5 | \( 1 + 3.16T + 5T^{2} \) |
| 11 | \( 1 + 6.32T + 11T^{2} \) |
| 13 | \( 1 + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.58 - 2.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 3.16T + 23T^{2} \) |
| 29 | \( 1 + (-1.58 - 2.73i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.74 + 8.21i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.74 + 8.21i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.74 + 8.21i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.32 - 10.9i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + (-2.5 + 4.33i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6 - 10.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.16 - 5.47i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (4.74 + 8.21i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.593387969760950106633315042855, −8.153718150284784794498646038871, −7.54239033466336510167096721357, −7.05783999090275954170411028413, −5.61237190613386970370411008740, −4.99947911295292348153478464319, −4.03960659843102050734638198724, −3.42146584452410823947183659146, −2.14922218074812560373109029108, −0.63830872377694276892959890316,
0.68965095114519755446956732667, 2.53434934434147814591895155631, 3.02762324538868913848914381860, 4.46981679914674845531551966966, 4.89822464012421767253703353747, 5.70444950146417962431354088647, 7.05478605714377526284808287545, 7.62263493301341788049229487336, 8.135415326988957778445567812612, 8.878881568054151945285699765153