| L(s) = 1 | + 1.93i·3-s + 1.24i·7-s − 0.747·9-s − 0.513·11-s − 6.15i·13-s − 4.51i·17-s + 19-s − 2.41·21-s + 5.86i·23-s + 4.36i·27-s − 6.62·29-s + 6.41·31-s − 0.994i·33-s − 1.40i·37-s + 11.9·39-s + ⋯ |
| L(s) = 1 | + 1.11i·3-s + 0.471i·7-s − 0.249·9-s − 0.154·11-s − 1.70i·13-s − 1.09i·17-s + 0.229·19-s − 0.526·21-s + 1.22i·23-s + 0.839i·27-s − 1.23·29-s + 1.15·31-s − 0.173i·33-s − 0.231i·37-s + 1.90·39-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
| \(L(1)\) |
\(\approx\) |
\(1.904790185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.904790185\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.93iT - 3T^{2} \) |
| 7 | \( 1 - 1.24iT - 7T^{2} \) |
| 11 | \( 1 + 0.513T + 11T^{2} \) |
| 13 | \( 1 + 6.15iT - 13T^{2} \) |
| 17 | \( 1 + 4.51iT - 17T^{2} \) |
| 23 | \( 1 - 5.86iT - 23T^{2} \) |
| 29 | \( 1 + 6.62T + 29T^{2} \) |
| 31 | \( 1 - 6.41T + 31T^{2} \) |
| 37 | \( 1 + 1.40iT - 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 3.04iT - 43T^{2} \) |
| 47 | \( 1 + 1.99iT - 47T^{2} \) |
| 53 | \( 1 - 14.0iT - 53T^{2} \) |
| 59 | \( 1 - 4.34T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 9.89iT - 67T^{2} \) |
| 71 | \( 1 - 7.42T + 71T^{2} \) |
| 73 | \( 1 + 12.8iT - 73T^{2} \) |
| 79 | \( 1 + 2.56T + 79T^{2} \) |
| 83 | \( 1 - 7.50iT - 83T^{2} \) |
| 89 | \( 1 - 7.85T + 89T^{2} \) |
| 97 | \( 1 - 6.74iT - 97T^{2} \) |
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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
−8.875500123198773167137604923680, −7.73266251676971299786305408133, −7.49165912272427043690306023550, −6.15859377639531834902946776570, −5.39801136566952672604348183905, −5.01823168677861334821308939458, −3.98088814146747056891007234543, −3.23143453283960679226685345615, −2.46473479051544597971760606207, −0.861521028665169785449872873799,
0.76900234455895775972323986886, 1.80311127535214692496464191956, 2.45758677886464496849698342143, 3.92009119293336901059967981539, 4.35047121023006779224318492429, 5.55374614754149416164197660689, 6.49294247031638952888690598555, 6.79766279186817661619693740256, 7.54464249077967711605222895146, 8.270454305515952307574163219281
