L(s) = 1 | + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)5-s + (−0.866 + 0.5i)7-s + 2i·11-s + (0.5 + 0.866i)13-s + (−0.866 − 0.499i)15-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (0.499 − 0.866i)21-s − i·27-s − 2·29-s + (−1 − 1.73i)33-s + (−0.866 − 0.499i)35-s − 37-s + (−0.866 − 0.499i)39-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)5-s + (−0.866 + 0.5i)7-s + 2i·11-s + (0.5 + 0.866i)13-s + (−0.866 − 0.499i)15-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (0.499 − 0.866i)21-s − i·27-s − 2·29-s + (−1 − 1.73i)33-s + (−0.866 − 0.499i)35-s − 37-s + (−0.866 − 0.499i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7299111879\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7299111879\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 - 2iT - T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 2T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.563060583749866065976975407476, −9.227277990226727676719224399743, −7.70703894349182852654001089648, −6.96098326942010593682754343903, −6.46641193038376928670583112525, −5.52937689194565315160023299661, −4.93657487872530910180363832398, −3.93115770749142610021613306321, −2.80247311086365203012754978867, −1.93413472847654885448222898557,
0.57599081996046052600021127654, 1.42250530226885003740679661387, 3.32328286489725431539489230532, 3.66430968086689078933731541613, 5.42403904498155839031749882017, 5.67895843105368293830297591377, 6.18000697797627914759832250468, 7.20355267944855152147036874962, 8.088418882388279938125382683533, 8.864512977739521613915007403203