| L(s) = 1 | − 2·3-s − 5-s + 3·7-s + 9-s + 3·11-s + 4·13-s + 2·15-s + 5·17-s − 6·21-s − 4·25-s + 4·27-s − 2·29-s + 8·31-s − 6·33-s − 3·35-s + 10·37-s − 8·39-s − 6·41-s + 7·43-s − 45-s + 9·47-s + 2·49-s − 10·51-s + 8·53-s − 3·55-s + 14·59-s − 5·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s + 0.904·11-s + 1.10·13-s + 0.516·15-s + 1.21·17-s − 1.30·21-s − 4/5·25-s + 0.769·27-s − 0.371·29-s + 1.43·31-s − 1.04·33-s − 0.507·35-s + 1.64·37-s − 1.28·39-s − 0.937·41-s + 1.06·43-s − 0.149·45-s + 1.31·47-s + 2/7·49-s − 1.40·51-s + 1.09·53-s − 0.404·55-s + 1.82·59-s − 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.672737517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.672737517\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−8.148060259363710086433322724847, −7.38601845396983265703380778176, −6.58341608756454848978175875806, −5.75508495262059909206416744964, −5.51991869338759908788687173901, −4.33693159410665846542938642673, −4.05153477392287557754730772390, −2.83555716155125599118268588753, −1.44011806260301581515132764673, −0.833864002051176659308197110715,
0.833864002051176659308197110715, 1.44011806260301581515132764673, 2.83555716155125599118268588753, 4.05153477392287557754730772390, 4.33693159410665846542938642673, 5.51991869338759908788687173901, 5.75508495262059909206416744964, 6.58341608756454848978175875806, 7.38601845396983265703380778176, 8.148060259363710086433322724847
