L(s) = 1 | + 1.25·3-s + 2.18i·5-s − 0.251i·7-s − 1.43·9-s + i·11-s + (3.07 + 1.88i)13-s + 2.73i·15-s + 5.77·17-s + 7.21i·19-s − 0.314i·21-s + 1.93·23-s + 0.234·25-s − 5.54·27-s − 5.64·29-s − 3.81i·31-s + ⋯ |
L(s) = 1 | + 0.722·3-s + 0.976i·5-s − 0.0949i·7-s − 0.478·9-s + 0.301i·11-s + (0.851 + 0.523i)13-s + 0.705i·15-s + 1.40·17-s + 1.65i·19-s − 0.0686i·21-s + 0.402·23-s + 0.0468·25-s − 1.06·27-s − 1.04·29-s − 0.685i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.523 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.56420 + 0.874494i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.56420 + 0.874494i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (-3.07 - 1.88i)T \) |
good | 3 | \( 1 - 1.25T + 3T^{2} \) |
| 5 | \( 1 - 2.18iT - 5T^{2} \) |
| 7 | \( 1 + 0.251iT - 7T^{2} \) |
| 17 | \( 1 - 5.77T + 17T^{2} \) |
| 19 | \( 1 - 7.21iT - 19T^{2} \) |
| 23 | \( 1 - 1.93T + 23T^{2} \) |
| 29 | \( 1 + 5.64T + 29T^{2} \) |
| 31 | \( 1 + 3.81iT - 31T^{2} \) |
| 37 | \( 1 - 6.68iT - 37T^{2} \) |
| 41 | \( 1 + 10.9iT - 41T^{2} \) |
| 43 | \( 1 + 7.64T + 43T^{2} \) |
| 47 | \( 1 - 6.50iT - 47T^{2} \) |
| 53 | \( 1 - 9.98T + 53T^{2} \) |
| 59 | \( 1 + 13.6iT - 59T^{2} \) |
| 61 | \( 1 + 9.64T + 61T^{2} \) |
| 67 | \( 1 + 11.7iT - 67T^{2} \) |
| 71 | \( 1 + 8.73iT - 71T^{2} \) |
| 73 | \( 1 - 11.9iT - 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 6.39iT - 83T^{2} \) |
| 89 | \( 1 + 6.82iT - 89T^{2} \) |
| 97 | \( 1 + 10.1iT - 97T^{2} \) |
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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
−10.78437634080783801715279964650, −10.01201216671627469904394198156, −9.127818592920123652064706831345, −8.130103810983504434721738194534, −7.47581113819197070910204405734, −6.36774714289204765774356299535, −5.50077292652449555561146087039, −3.79669096866697468628436107133, −3.20172384058522593607306585582, −1.84221215028483669303900888342,
1.03017185286626930310215862552, 2.74679088089360158355478735013, 3.71099589811184224271222357323, 5.09521446856649356401784199415, 5.78691567435165958869941776841, 7.19442194077791475795125475601, 8.212587385719606499652360097989, 8.796193044101980156019184319216, 9.386567541717340679474287184007, 10.59302379091554233794377308268