| L(s) = 1 | + 7-s + 13-s − 3·17-s − 2·19-s − 3·23-s + 3·29-s − 31-s − 2·37-s + 3·41-s + 7·43-s − 6·47-s + 49-s + 9·53-s + 3·59-s + 61-s − 8·67-s − 4·73-s + 8·79-s + 15·83-s − 6·89-s + 91-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.277·13-s − 0.727·17-s − 0.458·19-s − 0.625·23-s + 0.557·29-s − 0.179·31-s − 0.328·37-s + 0.468·41-s + 1.06·43-s − 0.875·47-s + 1/7·49-s + 1.23·53-s + 0.390·59-s + 0.128·61-s − 0.977·67-s − 0.468·73-s + 0.900·79-s + 1.64·83-s − 0.635·89-s + 0.104·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.89664256604981, −13.51285505201842, −13.12607269905402, −12.41888356132468, −12.11119812130235, −11.51254172043367, −11.05445484008377, −10.54127887895809, −10.21068121016929, −9.444939731852486, −9.022910254053754, −8.547880564875317, −7.972881159007220, −7.607041568284147, −6.780725680107427, −6.539353205254844, −5.807400359571680, −5.356305406028335, −4.658020075234098, −4.159459029739126, −3.703977881570720, −2.832016775143232, −2.301964835347468, −1.664338471754914, −0.8864427025874913, 0,
0.8864427025874913, 1.664338471754914, 2.301964835347468, 2.832016775143232, 3.703977881570720, 4.159459029739126, 4.658020075234098, 5.356305406028335, 5.807400359571680, 6.539353205254844, 6.780725680107427, 7.607041568284147, 7.972881159007220, 8.547880564875317, 9.022910254053754, 9.444939731852486, 10.21068121016929, 10.54127887895809, 11.05445484008377, 11.51254172043367, 12.11119812130235, 12.41888356132468, 13.12607269905402, 13.51285505201842, 13.89664256604981
