L(s) = 1 | − 3-s − 5-s − 3·7-s + 9-s − 11-s + 2·13-s + 15-s + 5·19-s + 3·21-s + 23-s + 25-s − 27-s − 8·29-s + 33-s + 3·35-s − 10·37-s − 2·39-s + 4·41-s − 43-s − 45-s − 8·47-s + 2·49-s − 53-s + 55-s − 5·57-s − 6·59-s + 2·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.258·15-s + 1.14·19-s + 0.654·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.48·29-s + 0.174·33-s + 0.507·35-s − 1.64·37-s − 0.320·39-s + 0.624·41-s − 0.152·43-s − 0.149·45-s − 1.16·47-s + 2/7·49-s − 0.137·53-s + 0.134·55-s − 0.662·57-s − 0.781·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 31 | \( 1 \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 17 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−13.00992600714236, −12.87745574156162, −12.12704728258309, −11.79792757173449, −11.34736550274862, −10.87884898512928, −10.37631842547375, −9.955684060213807, −9.430217299427247, −9.063288321376674, −8.492356599301892, −7.889560512217471, −7.216716873618937, −7.186215469661805, −6.388304666986556, −6.019762644678053, −5.479585267787276, −5.043431772912923, −4.394192632385246, −3.771654796804977, −3.229931816779420, −3.051800806842750, −1.992962604310201, −1.430826042331740, −0.5918496105317898, 0,
0.5918496105317898, 1.430826042331740, 1.992962604310201, 3.051800806842750, 3.229931816779420, 3.771654796804977, 4.394192632385246, 5.043431772912923, 5.479585267787276, 6.019762644678053, 6.388304666986556, 7.186215469661805, 7.216716873618937, 7.889560512217471, 8.492356599301892, 9.063288321376674, 9.430217299427247, 9.955684060213807, 10.37631842547375, 10.87884898512928, 11.34736550274862, 11.79792757173449, 12.12704728258309, 12.87745574156162, 13.00992600714236