| L(s) = 1 | + 2.82·3-s − i·5-s + 2.09i·7-s + 4.99·9-s + 1.73i·11-s − 2.82i·15-s + 3.62·17-s − 1.06i·19-s + 5.92i·21-s + 7.81·23-s − 25-s + 5.62·27-s − 0.526·29-s − 5.84i·31-s + 4.89i·33-s + ⋯ |
| L(s) = 1 | + 1.63·3-s − 0.447i·5-s + 0.791i·7-s + 1.66·9-s + 0.522i·11-s − 0.729i·15-s + 0.879·17-s − 0.245i·19-s + 1.29i·21-s + 1.63·23-s − 0.200·25-s + 1.08·27-s − 0.0978·29-s − 1.04i·31-s + 0.852i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.699070392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.699070392\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 7 | \( 1 - 2.09iT - 7T^{2} \) |
| 11 | \( 1 - 1.73iT - 11T^{2} \) |
| 17 | \( 1 - 3.62T + 17T^{2} \) |
| 19 | \( 1 + 1.06iT - 19T^{2} \) |
| 23 | \( 1 - 7.81T + 23T^{2} \) |
| 29 | \( 1 + 0.526T + 29T^{2} \) |
| 31 | \( 1 + 5.84iT - 31T^{2} \) |
| 37 | \( 1 - 9.74iT - 37T^{2} \) |
| 41 | \( 1 + 4.26iT - 41T^{2} \) |
| 43 | \( 1 + 9.34T + 43T^{2} \) |
| 47 | \( 1 - 3.46iT - 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 1.40iT - 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 10.8iT - 67T^{2} \) |
| 71 | \( 1 - 14.1iT - 71T^{2} \) |
| 73 | \( 1 + 2.64iT - 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 - 15.7iT - 83T^{2} \) |
| 89 | \( 1 + 5.52iT - 89T^{2} \) |
| 97 | \( 1 - 15.1iT - 97T^{2} \) |
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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
−8.687952425271133633355699194961, −8.052934451283700751563827620505, −7.41981365122733859151790730688, −6.60104164659693839082029784984, −5.45289208787585044548235666346, −4.75162944963213471602069850781, −3.76761346994668270174851062908, −2.96587880813309476487503032959, −2.27838989460491709124292579070, −1.25651092212177882352233756670,
1.05086459116767829827626299732, 2.16404612762182486822507371942, 3.32590329624150890591192653227, 3.38361694125636597608680694803, 4.49110552055094303831109209190, 5.51005525386630815360759373602, 6.65035443080033789390163836760, 7.36268908071707774545535317012, 7.74722676046300429683887190725, 8.715091348934781069750579516741
