L(s) = 1 | − 3-s − 2·7-s + 9-s − 2·13-s + 4·17-s + 6·19-s + 2·21-s − 27-s + 8·29-s − 8·31-s − 10·37-s + 2·39-s − 8·41-s − 2·43-s + 8·47-s − 3·49-s − 4·51-s + 2·53-s − 6·57-s + 12·59-s − 10·61-s − 2·63-s − 12·67-s + 8·71-s + 6·73-s + 2·79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.554·13-s + 0.970·17-s + 1.37·19-s + 0.436·21-s − 0.192·27-s + 1.48·29-s − 1.43·31-s − 1.64·37-s + 0.320·39-s − 1.24·41-s − 0.304·43-s + 1.16·47-s − 3/7·49-s − 0.560·51-s + 0.274·53-s − 0.794·57-s + 1.56·59-s − 1.28·61-s − 0.251·63-s − 1.46·67-s + 0.949·71-s + 0.702·73-s + 0.225·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.31970280728218, −14.61206007985636, −14.01014541618542, −13.68266400803999, −12.99882098774423, −12.34785758417019, −12.09736572502415, −11.67580517537190, −10.87815851902324, −10.26656687474574, −10.01874102265998, −9.386864776157856, −8.851606899279576, −8.115660212188827, −7.390991461700994, −7.076464797162382, −6.431268114970880, −5.786328049745563, −5.156750674967987, −4.892004684575666, −3.739587869258921, −3.407162494430413, −2.668122776055509, −1.702225038665370, −0.9049959723921419, 0,
0.9049959723921419, 1.702225038665370, 2.668122776055509, 3.407162494430413, 3.739587869258921, 4.892004684575666, 5.156750674967987, 5.786328049745563, 6.431268114970880, 7.076464797162382, 7.390991461700994, 8.115660212188827, 8.851606899279576, 9.386864776157856, 10.01874102265998, 10.26656687474574, 10.87815851902324, 11.67580517537190, 12.09736572502415, 12.34785758417019, 12.99882098774423, 13.68266400803999, 14.01014541618542, 14.61206007985636, 15.31970280728218