L(s) = 1 | + 5-s + 3·7-s − 2·13-s − 2·17-s + 19-s + 2·23-s + 25-s + 8·29-s + 5·31-s + 3·35-s − 11·37-s − 6·41-s − 8·43-s − 8·47-s + 2·49-s − 2·53-s − 4·59-s − 3·61-s − 2·65-s − 9·67-s + 6·71-s − 11·73-s − 9·79-s − 8·83-s − 2·85-s − 6·89-s − 6·91-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.13·7-s − 0.554·13-s − 0.485·17-s + 0.229·19-s + 0.417·23-s + 1/5·25-s + 1.48·29-s + 0.898·31-s + 0.507·35-s − 1.80·37-s − 0.937·41-s − 1.21·43-s − 1.16·47-s + 2/7·49-s − 0.274·53-s − 0.520·59-s − 0.384·61-s − 0.248·65-s − 1.09·67-s + 0.712·71-s − 1.28·73-s − 1.01·79-s − 0.878·83-s − 0.216·85-s − 0.635·89-s − 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−15.65501169184784, −15.31634279127026, −14.66480588156006, −14.17190387319874, −13.76543086956919, −13.21115907756830, −12.52145876344017, −11.83624452054918, −11.61821372144492, −10.84780206420789, −10.22023454649564, −9.950896977648980, −9.034121284691917, −8.510434558877831, −8.137922086599499, −7.311338900362441, −6.772997988795221, −6.197613308025778, −5.279821296279866, −4.858347345046910, −4.449122923761532, −3.324261824178293, −2.734763769981017, −1.792918180125112, −1.324120071984594, 0,
1.324120071984594, 1.792918180125112, 2.734763769981017, 3.324261824178293, 4.449122923761532, 4.858347345046910, 5.279821296279866, 6.197613308025778, 6.772997988795221, 7.311338900362441, 8.137922086599499, 8.510434558877831, 9.034121284691917, 9.950896977648980, 10.22023454649564, 10.84780206420789, 11.61821372144492, 11.83624452054918, 12.52145876344017, 13.21115907756830, 13.76543086956919, 14.17190387319874, 14.66480588156006, 15.31634279127026, 15.65501169184784