| L(s) = 1 | − 2-s − 2·3-s − 4-s + 3·5-s + 2·6-s − 4·7-s + 3·8-s + 9-s − 3·10-s + 4·11-s + 2·12-s + 2·13-s + 4·14-s − 6·15-s − 16-s + 3·17-s − 18-s − 19-s − 3·20-s + 8·21-s − 4·22-s + 7·23-s − 6·24-s + 4·25-s − 2·26-s + 4·27-s + 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 1.34·5-s + 0.816·6-s − 1.51·7-s + 1.06·8-s + 1/3·9-s − 0.948·10-s + 1.20·11-s + 0.577·12-s + 0.554·13-s + 1.06·14-s − 1.54·15-s − 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.229·19-s − 0.670·20-s + 1.74·21-s − 0.852·22-s + 1.45·23-s − 1.22·24-s + 4/5·25-s − 0.392·26-s + 0.769·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31939 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31939 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + T \) |
| 41 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.55014575348081, −14.55817991933032, −14.31944444351808, −13.58451102114650, −13.14592221952447, −12.70573050725389, −12.30489352964371, −11.40735908017762, −10.89137857400089, −10.48688962741210, −9.793930016818292, −9.503613269422172, −8.993601481517144, −8.708362808764819, −7.489171915720584, −7.008259593524127, −6.299678368647207, −6.082659321336762, −5.364591853536502, −5.001757809026971, −3.824501466370438, −3.533129843609955, −2.425201998509312, −1.436896363745034, −0.9175674879045381, 0,
0.9175674879045381, 1.436896363745034, 2.425201998509312, 3.533129843609955, 3.824501466370438, 5.001757809026971, 5.364591853536502, 6.082659321336762, 6.299678368647207, 7.008259593524127, 7.489171915720584, 8.708362808764819, 8.993601481517144, 9.503613269422172, 9.793930016818292, 10.48688962741210, 10.89137857400089, 11.40735908017762, 12.30489352964371, 12.70573050725389, 13.14592221952447, 13.58451102114650, 14.31944444351808, 14.55817991933032, 15.55014575348081
