L(s) = 1 | − 5-s − 11-s − 3·17-s − 4·19-s − 6·23-s + 25-s − 6·29-s − 7·31-s + 8·37-s + 8·41-s + 43-s + 2·47-s − 7·49-s + 12·53-s + 55-s + 3·59-s + 2·67-s − 8·71-s − 3·73-s − 8·79-s + 9·83-s + 3·85-s − 11·89-s + 4·95-s + 8·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.301·11-s − 0.727·17-s − 0.917·19-s − 1.25·23-s + 1/5·25-s − 1.11·29-s − 1.25·31-s + 1.31·37-s + 1.24·41-s + 0.152·43-s + 0.291·47-s − 49-s + 1.64·53-s + 0.134·55-s + 0.390·59-s + 0.244·67-s − 0.949·71-s − 0.351·73-s − 0.900·79-s + 0.987·83-s + 0.325·85-s − 1.16·89-s + 0.410·95-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.88296072866546, −12.47517697162411, −11.80637252971138, −11.44502604265131, −10.99678754229040, −10.68615837484667, −10.07536175726445, −9.654808228959270, −9.073724457948595, −8.750184825654707, −8.181004851387852, −7.725059954050977, −7.362419675095385, −6.826341968957307, −6.261616165950053, −5.743186334148395, −5.459708802518489, −4.595515929905403, −4.201524614034940, −3.909253966275782, −3.232035308029930, −2.478828833374619, −2.154863401192117, −1.502292838032422, −0.5809289559902394, 0,
0.5809289559902394, 1.502292838032422, 2.154863401192117, 2.478828833374619, 3.232035308029930, 3.909253966275782, 4.201524614034940, 4.595515929905403, 5.459708802518489, 5.743186334148395, 6.261616165950053, 6.826341968957307, 7.362419675095385, 7.725059954050977, 8.181004851387852, 8.750184825654707, 9.073724457948595, 9.654808228959270, 10.07536175726445, 10.68615837484667, 10.99678754229040, 11.44502604265131, 11.80637252971138, 12.47517697162411, 12.88296072866546