L(s) = 1 | + 2·5-s − 7-s − 3·9-s + 4·11-s − 6·17-s − 8·19-s − 25-s − 6·29-s + 8·31-s − 2·35-s − 2·37-s − 2·41-s − 4·43-s − 6·45-s − 8·47-s + 49-s − 6·53-s + 8·55-s + 6·61-s + 3·63-s + 4·67-s − 8·71-s − 10·73-s − 4·77-s − 16·79-s + 9·81-s − 8·83-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.377·7-s − 9-s + 1.20·11-s − 1.45·17-s − 1.83·19-s − 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.338·35-s − 0.328·37-s − 0.312·41-s − 0.609·43-s − 0.894·45-s − 1.16·47-s + 1/7·49-s − 0.824·53-s + 1.07·55-s + 0.768·61-s + 0.377·63-s + 0.488·67-s − 0.949·71-s − 1.17·73-s − 0.455·77-s − 1.80·79-s + 81-s − 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9006482330\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9006482330\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.06929234842114, −13.41792820351619, −13.22831441724580, −12.69854964168413, −11.99008811816699, −11.42331636105186, −11.24627035055101, −10.45461573590792, −10.04731260141984, −9.441928800920673, −8.974932202631965, −8.566843638996432, −8.194398793541814, −7.198447471395431, −6.599068927455776, −6.300438417881343, −5.957491663514909, −5.210301008323480, −4.470966572206251, −4.073841272954043, −3.285551625424022, −2.619801442249240, −1.991468591533222, −1.538260187510115, −0.2894098599452606,
0.2894098599452606, 1.538260187510115, 1.991468591533222, 2.619801442249240, 3.285551625424022, 4.073841272954043, 4.470966572206251, 5.210301008323480, 5.957491663514909, 6.300438417881343, 6.599068927455776, 7.198447471395431, 8.194398793541814, 8.566843638996432, 8.974932202631965, 9.441928800920673, 10.04731260141984, 10.45461573590792, 11.24627035055101, 11.42331636105186, 11.99008811816699, 12.69854964168413, 13.22831441724580, 13.41792820351619, 14.06929234842114