| L(s) = 1 | + i·3-s − 2.23i·5-s + i·7-s + 2·9-s + 2.23·11-s − 2.23i·13-s + 2.23·15-s + 2.23i·17-s + 4.47·19-s − 21-s + 4i·23-s − 5.00·25-s + 5i·27-s − 29-s + 2.23i·33-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.999i·5-s + 0.377i·7-s + 0.666·9-s + 0.674·11-s − 0.620i·13-s + 0.577·15-s + 0.542i·17-s + 1.02·19-s − 0.218·21-s + 0.834i·23-s − 1.00·25-s + 0.962i·27-s − 0.185·29-s + 0.389i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.038851964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.038851964\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 + 2.23iT - 13T^{2} \) |
| 17 | \( 1 - 2.23iT - 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 8.94iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 + 4.47iT - 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 - 13.4iT - 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 2.23iT - 97T^{2} \) |
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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
−9.215525314678675976603571987900, −8.372558814458001503166803416451, −7.60918676420151595381703289766, −6.72464491931959747639201900392, −5.46695701416177221568204135411, −5.27324172503916018765511062727, −4.04173861688421788630604594692, −3.59883459846303452851210336219, −2.03164135567164852220885381282, −0.943062176170527011755228258069,
1.02840239041770209080745946756, 2.13190167974315958213565080951, 3.19015667997840440688566057518, 4.08866089473425745504222418192, 4.99067705464786828848853516993, 6.38857888190956179183808445893, 6.61192655289090003929435140407, 7.42935400823676277060100054480, 7.968206362159959020026252842688, 9.169198762651226183820651722216
