L(s) = 1 | + 3·5-s + 2·7-s − 6·11-s + 5·13-s − 3·17-s + 2·19-s + 6·23-s + 4·25-s + 3·29-s − 4·31-s + 6·35-s + 5·37-s − 6·41-s − 10·43-s − 3·49-s − 6·53-s − 18·55-s − 12·59-s + 5·61-s + 15·65-s + 2·67-s + 6·71-s − 73-s − 12·77-s − 10·79-s − 9·85-s − 3·89-s + ⋯ |
L(s) = 1 | + 1.34·5-s + 0.755·7-s − 1.80·11-s + 1.38·13-s − 0.727·17-s + 0.458·19-s + 1.25·23-s + 4/5·25-s + 0.557·29-s − 0.718·31-s + 1.01·35-s + 0.821·37-s − 0.937·41-s − 1.52·43-s − 3/7·49-s − 0.824·53-s − 2.42·55-s − 1.56·59-s + 0.640·61-s + 1.86·65-s + 0.244·67-s + 0.712·71-s − 0.117·73-s − 1.36·77-s − 1.12·79-s − 0.976·85-s − 0.317·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.654176450\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.654176450\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−11.29217955372326396250803080771, −10.74003194403470753139037725219, −9.843664539048397564353364791948, −8.776320410384522753279292613425, −7.957203241521051791394470373598, −6.64268335409264911701355422086, −5.57441037932249871861950405953, −4.85946862051001909635264517777, −2.97036565258433729627708570907, −1.66955489747042459625131055657,
1.66955489747042459625131055657, 2.97036565258433729627708570907, 4.85946862051001909635264517777, 5.57441037932249871861950405953, 6.64268335409264911701355422086, 7.957203241521051791394470373598, 8.776320410384522753279292613425, 9.843664539048397564353364791948, 10.74003194403470753139037725219, 11.29217955372326396250803080771