| L(s) = 1 | + i·3-s − i·5-s − 2·7-s + 2·9-s + i·11-s + 4i·13-s + 15-s − 2·17-s − 2i·21-s − 23-s + 4·25-s + 5i·27-s − 7·31-s − 33-s + 2i·35-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447i·5-s − 0.755·7-s + 0.666·9-s + 0.301i·11-s + 1.10i·13-s + 0.258·15-s − 0.485·17-s − 0.436i·21-s − 0.208·23-s + 0.800·25-s + 0.962i·27-s − 1.25·31-s − 0.174·33-s + 0.338i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.031539122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.031539122\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - iT \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 5 | \( 1 + iT - 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + 3iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 5iT - 59T^{2} \) |
| 61 | \( 1 - 12iT - 61T^{2} \) |
| 67 | \( 1 - 7iT - 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−9.227180407775387332793625990020, −8.595995873790237691757765938325, −7.36375199317858547820177410741, −6.91679271442840710235842315596, −6.01538940583552023620483867352, −5.05663068126408195084243703201, −4.28360659033283494199441510909, −3.74543333508320484426818847308, −2.50457592797864243802896726666, −1.36747329116460545625661621007,
0.33422137679735750404242182948, 1.65194144038665596709273579850, 2.82555373629782158111136139760, 3.48600677096762046970341889300, 4.57760761337970861208606619946, 5.57615488775750965738337897656, 6.45533917905948251157907123640, 6.86060363043089862802918818442, 7.75164703125310646869963697921, 8.302527391626652276714107347508
