L(s) = 1 | + i·3-s + 2·4-s + 3i·7-s − 9-s − 4·11-s + 2i·12-s + 4·16-s + 3i·17-s + 8·19-s − 3·21-s + i·23-s − i·27-s + 6i·28-s − 9·29-s − 5·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 4-s + 1.13i·7-s − 0.333·9-s − 1.20·11-s + 0.577i·12-s + 16-s + 0.727i·17-s + 1.83·19-s − 0.654·21-s + 0.208i·23-s − 0.192i·27-s + 1.13i·28-s − 1.67·29-s − 0.898·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.814189703\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.814189703\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
good | 2 | \( 1 - 2T^{2} \) |
| 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 - 9iT - 37T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 - 13iT - 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 13iT - 67T^{2} \) |
| 71 | \( 1 + 13T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + iT - 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−9.565279338826413960971259546226, −8.926532559019562022816944927405, −7.75770461691618372062245578173, −7.50090320118022509772177881688, −6.00536206012827112974817969990, −5.71754463173331945510285075426, −4.83003050549640084082884850282, −3.36615658012461619407368530439, −2.78268081366154765488181893942, −1.69207891475095067150857139261,
0.63611834226249173720756556560, 1.90833694920186829033799034347, 2.91937636166616606508810148931, 3.79911427582326073073003096685, 5.25993559060611595006717698258, 5.77971687657674291872529010312, 7.06205038912928566350026728631, 7.42584380228740967621447704656, 7.77060578566631172700113499912, 9.100249813399176596182929467737