L(s) = 1 | + 2·5-s − 2.82i·7-s − 1.41i·11-s − 2.82i·13-s + 2·17-s + (1 − 4.24i)19-s − 1.41i·23-s − 25-s + 7.07i·29-s − 6·31-s − 5.65i·35-s − 11.3i·37-s + 4.24i·41-s − 7.07i·47-s − 1.00·49-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 1.06i·7-s − 0.426i·11-s − 0.784i·13-s + 0.485·17-s + (0.229 − 0.973i)19-s − 0.294i·23-s − 0.200·25-s + 1.31i·29-s − 1.07·31-s − 0.956i·35-s − 1.85i·37-s + 0.662i·41-s − 1.03i·47-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.827789511\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.827789511\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-1 + 4.24i)T \) |
good | 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 7.07iT - 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 11.3iT - 37T^{2} \) |
| 41 | \( 1 - 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 7.07iT - 47T^{2} \) |
| 53 | \( 1 - 9.89iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12.7iT - 83T^{2} \) |
| 89 | \( 1 + 9.89iT - 89T^{2} \) |
| 97 | \( 1 - 2.82iT - 97T^{2} \) |
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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
−8.803836303869388025887237990287, −7.59024009090100770350262745389, −7.28228606117241987537414687676, −6.21323297088349995759190368928, −5.57221263482427776957652132679, −4.77119336982004184635187421274, −3.70746400731045660772215764563, −2.90984757051743393826411315754, −1.69550106553071754436707702664, −0.56960718636903515176045558222,
1.58969387208120941012034376764, 2.21803184637689717829514482767, 3.29888196616525037243434017917, 4.38894172675062966291398448840, 5.36368366270603245986997134831, 5.91695232788834883211118404340, 6.56653721126262848569220390314, 7.61204689926753097539005080221, 8.314135296785240751337418251325, 9.254953751879335105640281244228