| 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
−9.895671435435661375052093302721, −8.917873448485738418976925036287, −8.042839552732844322877914196618, −7.51502674417526068996970297429, −6.21717837056333008368079013286, −5.92784283386793990284278201989, −4.65597475409727642130979728446, −3.69506006802044428832936526380, −2.72980270772981524649480466262, −1.49662067040026699176015477520,
0.52266399797388193265749536457, 1.99418399914460421491730417488, 3.19903527324773265687298610837, 4.26622178013737711065093141909, 5.08690864063991979921661330973, 5.97539100469090254141996378376, 6.92081370836361531459628249375, 7.72146833230911632654324762097, 8.591966587928575577382266202108, 9.197749076780096737488469830304
