L(s) = 1 | − 5-s − 2·7-s − 3·9-s − 4·11-s − 4·13-s + 6·17-s + 19-s − 2·23-s + 25-s − 6·29-s − 8·31-s + 2·35-s + 4·37-s + 6·41-s − 6·43-s + 3·45-s + 6·47-s − 3·49-s + 8·53-s + 4·55-s − 12·59-s + 6·61-s + 6·63-s + 4·65-s − 10·73-s + 8·77-s − 8·79-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.755·7-s − 9-s − 1.20·11-s − 1.10·13-s + 1.45·17-s + 0.229·19-s − 0.417·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.338·35-s + 0.657·37-s + 0.937·41-s − 0.914·43-s + 0.447·45-s + 0.875·47-s − 3/7·49-s + 1.09·53-s + 0.539·55-s − 1.56·59-s + 0.768·61-s + 0.755·63-s + 0.496·65-s − 1.17·73-s + 0.911·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−10.87957085208589286331284849801, −9.980294986705641550878798015674, −9.152851132600398164852591532224, −7.890968979292844991614448934934, −7.38646682250914023626660516961, −5.88641650006632163841711617715, −5.16746475704162288042683266443, −3.57032525010301463495530674648, −2.59335656559048749217684354708, 0,
2.59335656559048749217684354708, 3.57032525010301463495530674648, 5.16746475704162288042683266443, 5.88641650006632163841711617715, 7.38646682250914023626660516961, 7.890968979292844991614448934934, 9.152851132600398164852591532224, 9.980294986705641550878798015674, 10.87957085208589286331284849801