| L(s) = 1 | − 5-s + 7-s − 3·9-s − 11-s − 2·13-s + 2·17-s − 6·23-s + 25-s + 5·29-s − 5·31-s − 35-s + 6·37-s − 2·41-s + 8·43-s + 3·45-s − 4·47-s + 49-s + 55-s + 3·59-s − 9·61-s − 3·63-s + 2·65-s + 10·67-s − 3·71-s − 6·73-s − 77-s + 9·79-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 9-s − 0.301·11-s − 0.554·13-s + 0.485·17-s − 1.25·23-s + 1/5·25-s + 0.928·29-s − 0.898·31-s − 0.169·35-s + 0.986·37-s − 0.312·41-s + 1.21·43-s + 0.447·45-s − 0.583·47-s + 1/7·49-s + 0.134·55-s + 0.390·59-s − 1.15·61-s − 0.377·63-s + 0.248·65-s + 1.22·67-s − 0.356·71-s − 0.702·73-s − 0.113·77-s + 1.01·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9104828274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9104828274\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 17 T + p T^{2} \) |
| 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
−13.01080918980506, −12.39396769098536, −12.16661008960471, −11.67250875720836, −11.16927747485718, −10.84629434611572, −10.25790513601640, −9.720522043703754, −9.353045804503017, −8.620956509193057, −8.274136048199915, −7.891056257575977, −7.404072218027580, −6.896572403869890, −6.149005974948587, −5.793992666221470, −5.283858397808735, −4.721730160023230, −4.183300764479746, −3.654525728974977, −2.928645225344562, −2.570929353665226, −1.893737926636361, −1.102654132292852, −0.2860554027072842,
0.2860554027072842, 1.102654132292852, 1.893737926636361, 2.570929353665226, 2.928645225344562, 3.654525728974977, 4.183300764479746, 4.721730160023230, 5.283858397808735, 5.793992666221470, 6.149005974948587, 6.896572403869890, 7.404072218027580, 7.891056257575977, 8.274136048199915, 8.620956509193057, 9.353045804503017, 9.720522043703754, 10.25790513601640, 10.84629434611572, 11.16927747485718, 11.67250875720836, 12.16661008960471, 12.39396769098536, 13.01080918980506
