| L(s) = 1 | − 1.87i·3-s − 1.18i·7-s − 0.532·9-s − 2.18·11-s − 1.71i·13-s + 0.120i·17-s − 19-s − 2.22·21-s + 7.98i·23-s − 4.63i·27-s − 3.24·29-s − 8.41·31-s + 4.10i·33-s − 3.33i·37-s − 3.22·39-s + ⋯ |
| L(s) = 1 | − 1.08i·3-s − 0.447i·7-s − 0.177·9-s − 0.658·11-s − 0.476i·13-s + 0.0292i·17-s − 0.229·19-s − 0.485·21-s + 1.66i·23-s − 0.892i·27-s − 0.603·29-s − 1.51·31-s + 0.714i·33-s − 0.547i·37-s − 0.516·39-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
| \(L(1)\) |
\(\approx\) |
\(0.1270865811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1270865811\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.87iT - 3T^{2} \) |
| 7 | \( 1 + 1.18iT - 7T^{2} \) |
| 11 | \( 1 + 2.18T + 11T^{2} \) |
| 13 | \( 1 + 1.71iT - 13T^{2} \) |
| 17 | \( 1 - 0.120iT - 17T^{2} \) |
| 23 | \( 1 - 7.98iT - 23T^{2} \) |
| 29 | \( 1 + 3.24T + 29T^{2} \) |
| 31 | \( 1 + 8.41T + 31T^{2} \) |
| 37 | \( 1 + 3.33iT - 37T^{2} \) |
| 41 | \( 1 + 8.98T + 41T^{2} \) |
| 43 | \( 1 - 4.06iT - 43T^{2} \) |
| 47 | \( 1 - 1.71iT - 47T^{2} \) |
| 53 | \( 1 + 6.51iT - 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 6.53T + 61T^{2} \) |
| 67 | \( 1 - 2.18iT - 67T^{2} \) |
| 71 | \( 1 + 9.12T + 71T^{2} \) |
| 73 | \( 1 - 0.773iT - 73T^{2} \) |
| 79 | \( 1 + 1.63T + 79T^{2} \) |
| 83 | \( 1 - 2.44iT - 83T^{2} \) |
| 89 | \( 1 + 2.83T + 89T^{2} \) |
| 97 | \( 1 - 2.19iT - 97T^{2} \) |
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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
−7.68930783344109921200700207694, −7.43817485568618181143280263689, −6.70816999770833889191373815655, −5.77132493892717145213010733259, −5.20205424644797917296673119903, −4.03646227151723195465242112445, −3.22035561157570939371629440643, −2.09747928391501751434868957842, −1.32219762162599696935788915247, −0.03559923574072089856798902916,
1.79172157845969179923588481772, 2.78471797770305866376345666209, 3.71819956131482776892507752241, 4.47249596264246437902778712816, 5.12282393664470916913127980416, 5.85036417657808291843218844911, 6.79669963382393703493986802937, 7.52252832826002017083922685909, 8.569290555695769117846639102555, 8.942681669265276737220300684282
