L(s) = 1 | − i·5-s − 0.585i·7-s − 1.41·11-s + 2.24·13-s − 1.17i·17-s − 5.65i·19-s − 3.17·23-s − 25-s − 2i·29-s + 3.17i·31-s − 0.585·35-s + 4.58·37-s − 8.24i·41-s − 6.82i·43-s − 8·47-s + ⋯ |
L(s) = 1 | − 0.447i·5-s − 0.221i·7-s − 0.426·11-s + 0.621·13-s − 0.284i·17-s − 1.29i·19-s − 0.661·23-s − 0.200·25-s − 0.371i·29-s + 0.569i·31-s − 0.0990·35-s + 0.753·37-s − 1.28i·41-s − 1.04i·43-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.287094918\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.287094918\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
good | 7 | \( 1 + 0.585iT - 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 - 2.24T + 13T^{2} \) |
| 17 | \( 1 + 1.17iT - 17T^{2} \) |
| 19 | \( 1 + 5.65iT - 19T^{2} \) |
| 23 | \( 1 + 3.17T + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 - 3.17iT - 31T^{2} \) |
| 37 | \( 1 - 4.58T + 37T^{2} \) |
| 41 | \( 1 + 8.24iT - 41T^{2} \) |
| 43 | \( 1 + 6.82iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 6.82iT - 53T^{2} \) |
| 59 | \( 1 + 5.41T + 59T^{2} \) |
| 61 | \( 1 + 3.17T + 61T^{2} \) |
| 67 | \( 1 + 11.3iT - 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 0.828T + 73T^{2} \) |
| 79 | \( 1 - 6.48iT - 79T^{2} \) |
| 83 | \( 1 - 1.17T + 83T^{2} \) |
| 89 | \( 1 + 0.928iT - 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−9.197749076780096737488469830304, −8.591966587928575577382266202108, −7.72146833230911632654324762097, −6.92081370836361531459628249375, −5.97539100469090254141996378376, −5.08690864063991979921661330973, −4.26622178013737711065093141909, −3.19903527324773265687298610837, −1.99418399914460421491730417488, −0.52266399797388193265749536457,
1.49662067040026699176015477520, 2.72980270772981524649480466262, 3.69506006802044428832936526380, 4.65597475409727642130979728446, 5.92784283386793990284278201989, 6.21717837056333008368079013286, 7.51502674417526068996970297429, 8.042839552732844322877914196618, 8.917873448485738418976925036287, 9.895671435435661375052093302721