L(s) = 1 | − 5-s − 4·11-s − 2·13-s + 2·17-s − 4·19-s + 25-s − 10·29-s − 6·37-s − 6·41-s + 4·43-s − 8·47-s + 6·53-s + 4·55-s + 4·59-s − 10·61-s + 2·65-s − 4·67-s + 16·71-s + 14·73-s + 8·79-s + 4·83-s − 2·85-s + 10·89-s + 4·95-s − 10·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.20·11-s − 0.554·13-s + 0.485·17-s − 0.917·19-s + 1/5·25-s − 1.85·29-s − 0.986·37-s − 0.937·41-s + 0.609·43-s − 1.16·47-s + 0.824·53-s + 0.539·55-s + 0.520·59-s − 1.28·61-s + 0.248·65-s − 0.488·67-s + 1.89·71-s + 1.63·73-s + 0.900·79-s + 0.439·83-s − 0.216·85-s + 1.05·89-s + 0.410·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.61091909393483, −13.05550429397015, −12.74861672035111, −12.18278446963718, −11.86254136605327, −11.13147187601282, −10.70122486437883, −10.49459472178761, −9.647653692290048, −9.474147529062951, −8.693860722105408, −8.180215222420981, −7.849179847756566, −7.319464354196773, −6.843974110176879, −6.246934699053411, −5.530111536886670, −5.160371567982040, −4.721346099435654, −3.893094152749608, −3.536767359958356, −2.856383889212227, −2.165787910322566, −1.746724513270904, −0.6161420993904838, 0,
0.6161420993904838, 1.746724513270904, 2.165787910322566, 2.856383889212227, 3.536767359958356, 3.893094152749608, 4.721346099435654, 5.160371567982040, 5.530111536886670, 6.246934699053411, 6.843974110176879, 7.319464354196773, 7.849179847756566, 8.180215222420981, 8.693860722105408, 9.474147529062951, 9.647653692290048, 10.49459472178761, 10.70122486437883, 11.13147187601282, 11.86254136605327, 12.18278446963718, 12.74861672035111, 13.05550429397015, 13.61091909393483