| L(s) = 1 | + 2i·5-s + 7-s + 3·9-s − 4i·11-s − 2i·13-s − 6·17-s − 8i·19-s + 25-s − 6i·29-s + 8·31-s + 2i·35-s − 2i·37-s − 2·41-s − 4i·43-s + 6i·45-s + ⋯ |
| L(s) = 1 | + 0.894i·5-s + 0.377·7-s + 9-s − 1.20i·11-s − 0.554i·13-s − 1.45·17-s − 1.83i·19-s + 0.200·25-s − 1.11i·29-s + 1.43·31-s + 0.338i·35-s − 0.328i·37-s − 0.312·41-s − 0.609i·43-s + 0.894i·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.749096628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.749096628\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 - 2iT - 5T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 6iT - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.105361001425330309830059408909, −8.431007374193734365174550384293, −7.51503733376692892887126903313, −6.74282655436372727131412527977, −6.22118238633745572350094155708, −4.98023886211419191598620813075, −4.24523287725253576647584269256, −3.08945340157153625565987591156, −2.30882421588956529155062429466, −0.69607963127277611925559392862,
1.37384435999096037927494630262, 2.04685660685479934351914453189, 3.72728598893114482228370967194, 4.70044314759144490956577086640, 4.86400637365457365779841486142, 6.34838798950487487519238798343, 6.95009324022285998257289448568, 7.919749478658128228399276073598, 8.542412218189265106204449080903, 9.455564899697810592407247663845
