| L(s) = 1 | − 5-s − 4·7-s − 3·9-s − 2·13-s − 2·17-s − 4·19-s − 4·23-s + 25-s − 2·29-s + 8·31-s + 4·35-s − 6·37-s + 6·41-s + 8·43-s + 3·45-s − 4·47-s + 9·49-s − 6·53-s − 4·59-s − 2·61-s + 12·63-s + 2·65-s + 8·67-s + 6·73-s + 9·81-s + 16·83-s + 2·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 9-s − 0.554·13-s − 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.371·29-s + 1.43·31-s + 0.676·35-s − 0.986·37-s + 0.937·41-s + 1.21·43-s + 0.447·45-s − 0.583·47-s + 9/7·49-s − 0.824·53-s − 0.520·59-s − 0.256·61-s + 1.51·63-s + 0.248·65-s + 0.977·67-s + 0.702·73-s + 81-s + 1.75·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.27787429043229, −14.51413103349228, −13.98743926192081, −13.63002519236736, −12.82483891846893, −12.49732307375888, −12.11714029161833, −11.36266314009153, −10.96699432430593, −10.25415210831466, −9.849336946209800, −9.147238003860659, −8.840621701353260, −8.051876103574022, −7.680205257733274, −6.768685386478514, −6.423291104393401, −5.964428427358095, −5.222542440587419, −4.455191453946856, −3.874288246001712, −3.200997706621643, −2.654017386568744, −2.038489669302792, −0.6348491092840265, 0,
0.6348491092840265, 2.038489669302792, 2.654017386568744, 3.200997706621643, 3.874288246001712, 4.455191453946856, 5.222542440587419, 5.964428427358095, 6.423291104393401, 6.768685386478514, 7.680205257733274, 8.051876103574022, 8.840621701353260, 9.147238003860659, 9.849336946209800, 10.25415210831466, 10.96699432430593, 11.36266314009153, 12.11714029161833, 12.49732307375888, 12.82483891846893, 13.63002519236736, 13.98743926192081, 14.51413103349228, 15.27787429043229
