L(s) = 1 | + 3·5-s − 7-s + 3·11-s − 13-s + 6·17-s − 4·19-s − 3·23-s + 4·25-s + 3·29-s + 5·31-s − 3·35-s + 2·37-s + 3·41-s − 43-s − 9·47-s − 6·49-s − 6·53-s + 9·55-s − 3·59-s − 13·61-s − 3·65-s − 7·67-s − 12·71-s − 10·73-s − 3·77-s + 11·79-s − 9·83-s + ⋯ |
L(s) = 1 | + 1.34·5-s − 0.377·7-s + 0.904·11-s − 0.277·13-s + 1.45·17-s − 0.917·19-s − 0.625·23-s + 4/5·25-s + 0.557·29-s + 0.898·31-s − 0.507·35-s + 0.328·37-s + 0.468·41-s − 0.152·43-s − 1.31·47-s − 6/7·49-s − 0.824·53-s + 1.21·55-s − 0.390·59-s − 1.66·61-s − 0.372·65-s − 0.855·67-s − 1.42·71-s − 1.17·73-s − 0.341·77-s + 1.23·79-s − 0.987·83-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
\(L(1)\) |
\(\approx\) |
\(1.606758036\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.606758036\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−11.73021674707092477525032688054, −10.37565949482716114050516831861, −9.829100813571519672276680117258, −9.030469546776877959637256836224, −7.83158898312820695700656210982, −6.44480626867553575016189342242, −5.95294980865709062393257575560, −4.61062307661215217497617906734, −3.08086895658960916974773294043, −1.61468837604414932192324442564,
1.61468837604414932192324442564, 3.08086895658960916974773294043, 4.61062307661215217497617906734, 5.95294980865709062393257575560, 6.44480626867553575016189342242, 7.83158898312820695700656210982, 9.030469546776877959637256836224, 9.829100813571519672276680117258, 10.37565949482716114050516831861, 11.73021674707092477525032688054