| L(s) = 1 | + 2-s − 4-s − 3·8-s + 2·13-s − 16-s − 2·17-s − 6·19-s − 3·23-s + 2·26-s − 7·29-s − 2·31-s + 5·32-s − 2·34-s + 8·37-s − 6·38-s + 5·41-s − 7·43-s − 3·46-s − 2·52-s + 6·53-s − 7·58-s + 10·59-s − 7·61-s − 2·62-s + 7·64-s + 5·67-s + 2·68-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s + 0.554·13-s − 1/4·16-s − 0.485·17-s − 1.37·19-s − 0.625·23-s + 0.392·26-s − 1.29·29-s − 0.359·31-s + 0.883·32-s − 0.342·34-s + 1.31·37-s − 0.973·38-s + 0.780·41-s − 1.06·43-s − 0.442·46-s − 0.277·52-s + 0.824·53-s − 0.919·58-s + 1.30·59-s − 0.896·61-s − 0.254·62-s + 7/8·64-s + 0.610·67-s + 0.242·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.628020810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.628020810\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−16.61926358749564, −15.74483563433124, −15.20945036879121, −14.69081782046047, −14.30815422810820, −13.46199092239385, −13.11040563926872, −12.76398231623780, −11.95432036059858, −11.40127309615421, −10.78649577153838, −10.08838712665351, −9.369696018187384, −8.860456871585890, −8.285201277706972, −7.608009193149179, −6.646885593873036, −6.119125883668494, −5.551567360977953, −4.759136833407733, −4.062283761536627, −3.698303783140454, −2.660833641933034, −1.864649498893109, −0.5168829589861634,
0.5168829589861634, 1.864649498893109, 2.660833641933034, 3.698303783140454, 4.062283761536627, 4.759136833407733, 5.551567360977953, 6.119125883668494, 6.646885593873036, 7.608009193149179, 8.285201277706972, 8.860456871585890, 9.369696018187384, 10.08838712665351, 10.78649577153838, 11.40127309615421, 11.95432036059858, 12.76398231623780, 13.11040563926872, 13.46199092239385, 14.30815422810820, 14.69081782046047, 15.20945036879121, 15.74483563433124, 16.61926358749564
