| L(s) = 1 | − 2·7-s − 4·11-s + 2·13-s + 8·19-s + 6·23-s − 4·29-s − 4·31-s + 10·37-s + 10·41-s + 8·43-s − 6·47-s − 3·49-s − 2·53-s + 10·59-s − 4·61-s − 67-s + 8·71-s − 2·73-s + 8·77-s − 4·79-s − 4·83-s + 10·89-s − 4·91-s + 8·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.20·11-s + 0.554·13-s + 1.83·19-s + 1.25·23-s − 0.742·29-s − 0.718·31-s + 1.64·37-s + 1.56·41-s + 1.21·43-s − 0.875·47-s − 3/7·49-s − 0.274·53-s + 1.30·59-s − 0.512·61-s − 0.122·67-s + 0.949·71-s − 0.234·73-s + 0.911·77-s − 0.450·79-s − 0.439·83-s + 1.05·89-s − 0.419·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 241200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−12.99578499664058, −12.85830536757130, −12.36953158620036, −11.49737551116562, −11.30484308029535, −10.90634306744368, −10.31156460852684, −9.781073025977080, −9.396120428318100, −9.098402200626674, −8.425921935161946, −7.782977815465624, −7.444732060038383, −7.183305948284333, −6.280300747545333, −6.015329308697387, −5.408662011310984, −5.033477361632557, −4.438000644803762, −3.595803961483992, −3.381421113023807, −2.641547387527838, −2.374648791332855, −1.259340163636302, −0.8545525888850270, 0,
0.8545525888850270, 1.259340163636302, 2.374648791332855, 2.641547387527838, 3.381421113023807, 3.595803961483992, 4.438000644803762, 5.033477361632557, 5.408662011310984, 6.015329308697387, 6.280300747545333, 7.183305948284333, 7.444732060038383, 7.782977815465624, 8.425921935161946, 9.098402200626674, 9.396120428318100, 9.781073025977080, 10.31156460852684, 10.90634306744368, 11.30484308029535, 11.49737551116562, 12.36953158620036, 12.85830536757130, 12.99578499664058
