L(s) = 1 | − 2i·5-s + 2·7-s + 4i·11-s − 2i·13-s − 4·17-s − 4i·19-s + 8·23-s + 25-s + 6i·29-s + 6·31-s − 4i·35-s − 2i·37-s + 12·41-s − 12i·43-s − 8·47-s + ⋯ |
L(s) = 1 | − 0.894i·5-s + 0.755·7-s + 1.20i·11-s − 0.554i·13-s − 0.970·17-s − 0.917i·19-s + 1.66·23-s + 0.200·25-s + 1.11i·29-s + 1.07·31-s − 0.676i·35-s − 0.328i·37-s + 1.87·41-s − 1.82i·43-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.954215940\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.954215940\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.911966787337030074926601960844, −8.291075298961237783859183623115, −7.28633607395926117884851400993, −6.81602000442399146212800856492, −5.50937409345539298850416806282, −4.75462772857192464273749896097, −4.47799024983447709974739020709, −2.98823899997234689004799977554, −1.92630733709849888654509153709, −0.814333099439137324094436879149,
1.10692434359277039338905228255, 2.44386756708720472514946110427, 3.21736699297368596350936873975, 4.28937179762605532913910805486, 5.07877775972606713572339068965, 6.29062222874349048401808275412, 6.52753032453984109426690266289, 7.70541239741409786635088172892, 8.218781264440375284238641931958, 9.071770219641167348781814342603