| L(s) = 1 | − 2·2-s + 4·4-s − 5i·5-s + 17.5·7-s − 8·8-s + 10i·10-s − 49.3i·11-s − 39.2i·13-s − 35.0·14-s + 16·16-s − 94.2i·17-s + (44.9 − 69.5i)19-s − 20i·20-s + 98.7i·22-s − 185. i·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447i·5-s + 0.946·7-s − 0.353·8-s + 0.316i·10-s − 1.35i·11-s − 0.836i·13-s − 0.669·14-s + 0.250·16-s − 1.34i·17-s + (0.542 − 0.840i)19-s − 0.223i·20-s + 0.956i·22-s − 1.68i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.372i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.927 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.416894463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.416894463\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5iT \) |
| 19 | \( 1 + (-44.9 + 69.5i)T \) |
| good | 7 | \( 1 - 17.5T + 343T^{2} \) |
| 11 | \( 1 + 49.3iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 39.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 94.2iT - 4.91e3T^{2} \) |
| 23 | \( 1 + 185. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 78.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 28.3iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 0.416iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 67.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 254.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 69.4iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 262.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 203.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 47.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 822. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 661.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 344.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 702. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 237. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 808.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.79e3iT - 9.12e5T^{2} \) |
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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
−8.479590631856576774239524313051, −8.147384738848446155191336782973, −7.20768618379846606134909676158, −6.29361997912104232230252351089, −5.28573463683921910487352701147, −4.70562279665244624084135018690, −3.26117808386790643844825414740, −2.43423819535215067836659926394, −1.00590093247399193857897664934, −0.43484984687100558841430908769,
1.61308578915559657180781774055, 1.80983785044221856473978607821, 3.34463480303733734010997879847, 4.32646148827580259711547749831, 5.29155017538104009344257888097, 6.31232763725457584152126783238, 7.11794741868337012793649859588, 7.81110694407464440705718906866, 8.386117730517789047497171542218, 9.500023881665088438276210377519
