| L(s) = 1 | + 0.496i·3-s − i·5-s − 1.35·7-s + 2.75·9-s − 5.74i·11-s − 7.05i·13-s + 0.496·15-s − 1.74·17-s − i·19-s − 0.671i·21-s − 0.0940·23-s − 25-s + 2.85i·27-s + 6.87i·29-s − 5.35·31-s + ⋯ |
| L(s) = 1 | + 0.286i·3-s − 0.447i·5-s − 0.511·7-s + 0.917·9-s − 1.73i·11-s − 1.95i·13-s + 0.128·15-s − 0.423·17-s − 0.229i·19-s − 0.146i·21-s − 0.0196·23-s − 0.200·25-s + 0.549i·27-s + 1.27i·29-s − 0.962·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 + 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 + 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9652031177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9652031177\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 19 | \( 1 + iT \) |
| good | 3 | \( 1 - 0.496iT - 3T^{2} \) |
| 7 | \( 1 + 1.35T + 7T^{2} \) |
| 11 | \( 1 + 5.74iT - 11T^{2} \) |
| 13 | \( 1 + 7.05iT - 13T^{2} \) |
| 17 | \( 1 + 1.74T + 17T^{2} \) |
| 23 | \( 1 + 0.0940T + 23T^{2} \) |
| 29 | \( 1 - 6.87iT - 29T^{2} \) |
| 31 | \( 1 + 5.35T + 31T^{2} \) |
| 37 | \( 1 + 3.03iT - 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 - 2.45iT - 43T^{2} \) |
| 47 | \( 1 - 0.916T + 47T^{2} \) |
| 53 | \( 1 - 5.92iT - 53T^{2} \) |
| 59 | \( 1 - 12.3iT - 59T^{2} \) |
| 61 | \( 1 - 11.8iT - 61T^{2} \) |
| 67 | \( 1 + 6.31iT - 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 6.72T + 79T^{2} \) |
| 83 | \( 1 + 17.9iT - 83T^{2} \) |
| 89 | \( 1 - 4.99T + 89T^{2} \) |
| 97 | \( 1 + 9.61T + 97T^{2} \) |
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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
−8.588175718852576303184653570235, −7.69314544638651085572453547110, −6.95617420961123914223765585734, −5.89563030062499920532799179660, −5.46435203506626533917126176821, −4.51796578560390359528756050265, −3.42332975024635741314008082028, −3.03538742893014921733554472395, −1.38426322890929111964750739562, −0.29634173332866620875347153322,
1.79106219336287187725910823881, 2.11136511169959825366114827787, 3.63746193366031336039986652532, 4.32237627970635443574212095741, 5.00839896473497403880890423044, 6.46423027006615656586671112331, 6.74531614634667395476634053135, 7.28185428740206453174166287268, 8.166478960666273989453150145493, 9.278781703405510561572352388931
