| L(s) = 1 | + 1.53i·3-s − 2.22i·7-s + 0.652·9-s − 3.22·11-s − 1.57i·13-s + 3.53i·17-s − 19-s + 3.41·21-s − 4.47i·23-s + 5.59i·27-s − 1.92·29-s − 3.81·31-s − 4.94i·33-s + 11.3i·37-s + 2.41·39-s + ⋯ |
| L(s) = 1 | + 0.884i·3-s − 0.841i·7-s + 0.217·9-s − 0.972·11-s − 0.436i·13-s + 0.856i·17-s − 0.229·19-s + 0.744·21-s − 0.933i·23-s + 1.07i·27-s − 0.356·29-s − 0.685·31-s − 0.860i·33-s + 1.86i·37-s + 0.386·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.264017469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.264017469\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.53iT - 3T^{2} \) |
| 7 | \( 1 + 2.22iT - 7T^{2} \) |
| 11 | \( 1 + 3.22T + 11T^{2} \) |
| 13 | \( 1 + 1.57iT - 13T^{2} \) |
| 17 | \( 1 - 3.53iT - 17T^{2} \) |
| 23 | \( 1 + 4.47iT - 23T^{2} \) |
| 29 | \( 1 + 1.92T + 29T^{2} \) |
| 31 | \( 1 + 3.81T + 31T^{2} \) |
| 37 | \( 1 - 11.3iT - 37T^{2} \) |
| 41 | \( 1 - 3.47T + 41T^{2} \) |
| 43 | \( 1 - 1.69iT - 43T^{2} \) |
| 47 | \( 1 - 1.57iT - 47T^{2} \) |
| 53 | \( 1 - 7.12iT - 53T^{2} \) |
| 59 | \( 1 - 7.88T + 59T^{2} \) |
| 61 | \( 1 + 2.79T + 61T^{2} \) |
| 67 | \( 1 - 3.22iT - 67T^{2} \) |
| 71 | \( 1 + 4.38T + 71T^{2} \) |
| 73 | \( 1 - 6.41iT - 73T^{2} \) |
| 79 | \( 1 - 8.59T + 79T^{2} \) |
| 83 | \( 1 - 14.6iT - 83T^{2} \) |
| 89 | \( 1 + 6.10T + 89T^{2} \) |
| 97 | \( 1 - 15.7iT - 97T^{2} \) |
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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
−8.710907953743594548335693663638, −8.030568300854238155044246102527, −7.34930903687002145175044611328, −6.53537049771155399373271033317, −5.61538747500184764903130206185, −4.82054973855726744032615481180, −4.18556044066584892034250921701, −3.47444658216284777387716226444, −2.46660618482711564104880725569, −1.12102964089180988767236663900,
0.39252220876828987987954635959, 1.84908699327760012402440808353, 2.37141159328834063770073817856, 3.45748708349848142013525935853, 4.54290047917719340508368898619, 5.47046918310414456331123327965, 5.94015549345399367899085072756, 7.05338695359205838625309327310, 7.37992334402623379471424678195, 8.129276986142996738183219079503
