| L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s + 3i·7-s − i·8-s − 9-s − 3·11-s + i·12-s − 4i·13-s − 3·14-s + 16-s + i·17-s − i·18-s + 5·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s + 1.13i·7-s − 0.353i·8-s − 0.333·9-s − 0.904·11-s + 0.288i·12-s − 1.10i·13-s − 0.801·14-s + 0.250·16-s + 0.242i·17-s − 0.235i·18-s + 1.14·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.472512896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.472512896\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 - iT \) |
| good | 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 - 3iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 - 11iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 13iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.711810076340587079370358538648, −8.172970612614319067779007494073, −7.62880002683200570807575205675, −6.72720772258649592744899043674, −5.81024039913179164477572307631, −5.47729904627234782254924807134, −4.55517624917281616818850059403, −3.08724985745249765195677137774, −2.49832566884053435184392029293, −0.910388054158702927031553283718,
0.65194847193031442096880134526, 1.97532343030252417689562816126, 3.14826157351925821323774196468, 3.84319716434560564264625713413, 4.69633561241729804880854703105, 5.29907367924317703658930503650, 6.46549599912781176754928565284, 7.41631173451711677372486432169, 7.994025721984093618081686782111, 9.063984090501083176599876361603
