| L(s) = 1 | + 2.28i·3-s − 2.23·9-s − 5.47·11-s − 0.874i·13-s − 4.57i·17-s − 5.99·19-s − 3.47i·23-s + 1.74i·27-s − 0.236·29-s + 8.27·31-s − 12.5i·33-s − 4.23i·37-s + 2·39-s + 5.11·41-s + 3.76i·43-s + ⋯ |
| L(s) = 1 | + 1.32i·3-s − 0.745·9-s − 1.64·11-s − 0.242i·13-s − 1.10i·17-s − 1.37·19-s − 0.723i·23-s + 0.336i·27-s − 0.0438·29-s + 1.48·31-s − 2.17i·33-s − 0.696i·37-s + 0.320·39-s + 0.799·41-s + 0.573i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.353122237\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.353122237\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.28iT - 3T^{2} \) |
| 11 | \( 1 + 5.47T + 11T^{2} \) |
| 13 | \( 1 + 0.874iT - 13T^{2} \) |
| 17 | \( 1 + 4.57iT - 17T^{2} \) |
| 19 | \( 1 + 5.99T + 19T^{2} \) |
| 23 | \( 1 + 3.47iT - 23T^{2} \) |
| 29 | \( 1 + 0.236T + 29T^{2} \) |
| 31 | \( 1 - 8.27T + 31T^{2} \) |
| 37 | \( 1 + 4.23iT - 37T^{2} \) |
| 41 | \( 1 - 5.11T + 41T^{2} \) |
| 43 | \( 1 - 3.76iT - 43T^{2} \) |
| 47 | \( 1 + 4.91iT - 47T^{2} \) |
| 53 | \( 1 - 11.7iT - 53T^{2} \) |
| 59 | \( 1 - 1.95T + 59T^{2} \) |
| 61 | \( 1 - 7.53T + 61T^{2} \) |
| 67 | \( 1 - 13.9iT - 67T^{2} \) |
| 71 | \( 1 - 16.7T + 71T^{2} \) |
| 73 | \( 1 + 7.53iT - 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 12.1iT - 83T^{2} \) |
| 89 | \( 1 + 5.86T + 89T^{2} \) |
| 97 | \( 1 + 16.3iT - 97T^{2} \) |
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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
−8.428065377562205822510388671658, −7.73750895716604558040638011553, −6.88841244059735077901961243110, −5.93069420442538481285318536422, −5.18148471329829408288693028290, −4.63428330888575711503866193027, −3.99616470945853976467623327249, −2.87836767080818095722132930265, −2.39667108098327590552879387369, −0.50743589074600627076114100788,
0.74630754371357451286943424663, 1.97464729928448389761696059499, 2.42166350431996341884067691103, 3.56297430102565989101202497344, 4.59786130508674770343915341837, 5.41797278677712374240062289687, 6.31140467331692787418685436006, 6.65259299864858846924781701017, 7.64099346998034946267355651960, 8.099129658555408359509400836840
