L(s) = 1 | + i·2-s − 4-s + (1.42 + 1.72i)5-s + 4.14i·7-s − i·8-s + (−1.72 + 1.42i)10-s + 4.76·11-s − 3.91i·13-s − 4.14·14-s + 16-s + 3.31i·17-s + 1.85·19-s + (−1.42 − 1.72i)20-s + 4.76i·22-s + 1.54i·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.635 + 0.771i)5-s + 1.56i·7-s − 0.353i·8-s + (−0.545 + 0.449i)10-s + 1.43·11-s − 1.08i·13-s − 1.10·14-s + 0.250·16-s + 0.804i·17-s + 0.424·19-s + (−0.317 − 0.385i)20-s + 1.01i·22-s + 0.321i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.635 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.358486383\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.358486383\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.42 - 1.72i)T \) |
| 37 | \( 1 + iT \) |
good | 7 | \( 1 - 4.14iT - 7T^{2} \) |
| 11 | \( 1 - 4.76T + 11T^{2} \) |
| 13 | \( 1 + 3.91iT - 13T^{2} \) |
| 17 | \( 1 - 3.31iT - 17T^{2} \) |
| 19 | \( 1 - 1.85T + 19T^{2} \) |
| 23 | \( 1 - 1.54iT - 23T^{2} \) |
| 29 | \( 1 - 8.87T + 29T^{2} \) |
| 31 | \( 1 - 9.75T + 31T^{2} \) |
| 41 | \( 1 - 5.06T + 41T^{2} \) |
| 43 | \( 1 + 9.99iT - 43T^{2} \) |
| 47 | \( 1 - 4.82iT - 47T^{2} \) |
| 53 | \( 1 - 5.13iT - 53T^{2} \) |
| 59 | \( 1 + 1.05T + 59T^{2} \) |
| 61 | \( 1 + 4.14T + 61T^{2} \) |
| 67 | \( 1 + 1.00iT - 67T^{2} \) |
| 71 | \( 1 + 6.45T + 71T^{2} \) |
| 73 | \( 1 - 10.7iT - 73T^{2} \) |
| 79 | \( 1 - 1.19T + 79T^{2} \) |
| 83 | \( 1 + 10.6iT - 83T^{2} \) |
| 89 | \( 1 - 7.29T + 89T^{2} \) |
| 97 | \( 1 + 14.8iT - 97T^{2} \) |
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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
−8.770929670824646468710427212657, −8.276181017638129446114877702929, −7.30105368414322569452435634130, −6.39790218430883706535876998899, −6.03736185110120042531011251209, −5.43861162694044109206614463451, −4.39936229919781119786176050573, −3.24862767531911454045385777226, −2.55567826979817689293099061639, −1.29808331417674363094933237587,
0.899550187736010970057749960484, 1.31677006128668609160918847260, 2.61260410396954960651581222428, 3.76887170234604133326437684507, 4.50516761700237652742533831356, 4.83079624532740724585522102266, 6.35924057039972200156781574148, 6.64556620419456020555228713400, 7.71286407808177976568476818670, 8.569256235356302798450741542066