| L(s) = 1 | + 0.874i·3-s + 2.23·5-s − 2.82i·7-s + 2.23·9-s − 0.763·11-s − 5.45i·13-s + 1.95i·15-s + 7.40i·17-s + 19-s + 2.47·21-s − 1.08i·23-s + 5.00·25-s + 4.57i·27-s + 4.47·29-s − 4·31-s + ⋯ |
| L(s) = 1 | + 0.504i·3-s + 0.999·5-s − 1.06i·7-s + 0.745·9-s − 0.230·11-s − 1.51i·13-s + 0.504i·15-s + 1.79i·17-s + 0.229·19-s + 0.539·21-s − 0.225i·23-s + 1.00·25-s + 0.880i·27-s + 0.830·29-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.65138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.65138\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.874iT - 3T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 + 5.45iT - 13T^{2} \) |
| 17 | \( 1 - 7.40iT - 17T^{2} \) |
| 23 | \( 1 + 1.08iT - 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2.62iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 8.48iT - 47T^{2} \) |
| 53 | \( 1 - 2.62iT - 53T^{2} \) |
| 59 | \( 1 - 1.52T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 11.1iT - 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 5.24iT - 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 - 13.7iT - 83T^{2} \) |
| 89 | \( 1 + 2.94T + 89T^{2} \) |
| 97 | \( 1 + 13.9iT - 97T^{2} \) |
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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
−10.87015284721003663439321662361, −10.28630654483392902719589071969, −9.971381667600476829510283835129, −8.623711540590768860086097545898, −7.62948553413514924175436173275, −6.52667752527951512050078391828, −5.49154529873637549125296879108, −4.39665493458959059464216497439, −3.23214929709971508899963411252, −1.42390425852460577838482783963,
1.69017227416043172698776581242, 2.70591854902366443509550573097, 4.61594104495603038059802412130, 5.60233128917662150501563586196, 6.66240158823599218079173816162, 7.37079396287498764148208787598, 8.903418515690833487855295929359, 9.353439188083905961992883264510, 10.29983724142265423688921356704, 11.62978753749777625894881876010
