| L(s) = 1 | + 1.60·5-s − 0.181i·7-s + 2.05i·11-s − 0.949i·13-s + 3.83i·17-s + 6.15i·19-s + 4.73·23-s − 2.42·25-s + 3.26i·29-s − 7.12·31-s − 0.291i·35-s − 4.42·37-s + (0.372 − 6.39i)41-s − 7.54·43-s + 10.2i·47-s + ⋯ |
| L(s) = 1 | + 0.717·5-s − 0.0687i·7-s + 0.618i·11-s − 0.263i·13-s + 0.930i·17-s + 1.41i·19-s + 0.987·23-s − 0.484·25-s + 0.605i·29-s − 1.28·31-s − 0.0493i·35-s − 0.727·37-s + (0.0582 − 0.998i)41-s − 1.15·43-s + 1.49i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0582 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0582 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.689830502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.689830502\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 41 | \( 1 + (-0.372 + 6.39i)T \) |
| good | 5 | \( 1 - 1.60T + 5T^{2} \) |
| 7 | \( 1 + 0.181iT - 7T^{2} \) |
| 11 | \( 1 - 2.05iT - 11T^{2} \) |
| 13 | \( 1 + 0.949iT - 13T^{2} \) |
| 17 | \( 1 - 3.83iT - 17T^{2} \) |
| 19 | \( 1 - 6.15iT - 19T^{2} \) |
| 23 | \( 1 - 4.73T + 23T^{2} \) |
| 29 | \( 1 - 3.26iT - 29T^{2} \) |
| 31 | \( 1 + 7.12T + 31T^{2} \) |
| 37 | \( 1 + 4.42T + 37T^{2} \) |
| 43 | \( 1 + 7.54T + 43T^{2} \) |
| 47 | \( 1 - 10.2iT - 47T^{2} \) |
| 53 | \( 1 - 0.554iT - 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + 9.80T + 61T^{2} \) |
| 67 | \( 1 - 7.91iT - 67T^{2} \) |
| 71 | \( 1 - 3.73iT - 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 8.84iT - 79T^{2} \) |
| 83 | \( 1 - 1.70T + 83T^{2} \) |
| 89 | \( 1 - 4.76iT - 89T^{2} \) |
| 97 | \( 1 + 9.39iT - 97T^{2} \) |
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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
−8.968720457608307240040229669649, −8.207073894096243505073139921294, −7.38821520147804814871590982405, −6.67613592817155349022040418563, −5.73829288538697141505940267409, −5.30763948230355957160423278126, −4.14137429206870761199712459216, −3.39042608811034907437630557455, −2.14981895577315241833913710910, −1.40268589954349397974121017253,
0.52066281085968050218160781695, 1.89260644021547095338657760799, 2.79364610578420736463928388320, 3.69990777061748047402783835077, 4.92911506209472018646943843468, 5.35658736002669903451218926146, 6.36977612389958824043747592158, 6.95953422222405964965002127268, 7.76628801802387251352102866586, 8.850463946504120442320849357879
