| L(s) = 1 | − 7-s − 3·9-s + 4·11-s − 6·13-s − 2·17-s − 6·29-s − 8·31-s − 10·37-s + 2·41-s − 4·43-s + 8·47-s + 49-s − 2·53-s − 8·59-s + 14·61-s + 3·63-s + 12·67-s + 16·71-s − 2·73-s − 4·77-s + 8·79-s + 9·81-s − 8·83-s + 10·89-s + 6·91-s − 2·97-s − 12·99-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s + 1.20·11-s − 1.66·13-s − 0.485·17-s − 1.11·29-s − 1.43·31-s − 1.64·37-s + 0.312·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.274·53-s − 1.04·59-s + 1.79·61-s + 0.377·63-s + 1.46·67-s + 1.89·71-s − 0.234·73-s − 0.455·77-s + 0.900·79-s + 81-s − 0.878·83-s + 1.05·89-s + 0.628·91-s − 0.203·97-s − 1.20·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9912922790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9912922790\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.73372781801255, −15.88930639178593, −15.27684321780343, −14.56306399657711, −14.39782157083042, −13.76416291022962, −12.94318945912598, −12.38840992089542, −11.92980874993171, −11.30404182037826, −10.78275699823388, −9.951186260354586, −9.298277293803813, −9.047539951711544, −8.251611232435457, −7.416612639710277, −6.922528072022910, −6.307683576973520, −5.415872288776436, −5.049059885035755, −3.938994409757820, −3.482844279846286, −2.466261182628301, −1.862680866993682, −0.4355700756094492,
0.4355700756094492, 1.862680866993682, 2.466261182628301, 3.482844279846286, 3.938994409757820, 5.049059885035755, 5.415872288776436, 6.307683576973520, 6.922528072022910, 7.416612639710277, 8.251611232435457, 9.047539951711544, 9.298277293803813, 9.951186260354586, 10.78275699823388, 11.30404182037826, 11.92980874993171, 12.38840992089542, 12.94318945912598, 13.76416291022962, 14.39782157083042, 14.56306399657711, 15.27684321780343, 15.88930639178593, 16.73372781801255
