L(s) = 1 | − 3-s − 3·7-s + 9-s + 11-s − 13-s − 5·17-s + 8·19-s + 3·21-s − 27-s − 29-s + 3·31-s − 33-s + 8·37-s + 39-s − 2·41-s − 8·43-s − 11·47-s + 2·49-s + 5·51-s + 11·53-s − 8·57-s − 5·59-s − 61-s − 3·63-s − 3·67-s + 16·71-s − 4·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s − 1.21·17-s + 1.83·19-s + 0.654·21-s − 0.192·27-s − 0.185·29-s + 0.538·31-s − 0.174·33-s + 1.31·37-s + 0.160·39-s − 0.312·41-s − 1.21·43-s − 1.60·47-s + 2/7·49-s + 0.700·51-s + 1.51·53-s − 1.05·57-s − 0.650·59-s − 0.128·61-s − 0.377·63-s − 0.366·67-s + 1.89·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−14.56677479304111, −13.81517179214112, −13.38562078737937, −13.16318243907473, −12.39612647955220, −12.03123366657443, −11.44562669406362, −11.15760666094344, −10.35564422447067, −9.904207275771850, −9.423288006471314, −9.164218164748209, −8.238603561217949, −7.781582525058372, −7.011264871286580, −6.618691867548970, −6.298227233741467, −5.471819030469926, −5.071631672620880, −4.372469473024091, −3.715796166030284, −3.114959670814462, −2.524286064657432, −1.607785814138116, −0.7902111168218075, 0,
0.7902111168218075, 1.607785814138116, 2.524286064657432, 3.114959670814462, 3.715796166030284, 4.372469473024091, 5.071631672620880, 5.471819030469926, 6.298227233741467, 6.618691867548970, 7.011264871286580, 7.781582525058372, 8.238603561217949, 9.164218164748209, 9.423288006471314, 9.904207275771850, 10.35564422447067, 11.15760666094344, 11.44562669406362, 12.03123366657443, 12.39612647955220, 13.16318243907473, 13.38562078737937, 13.81517179214112, 14.56677479304111