L(s) = 1 | − 5.61i·3-s − 7.03·7-s − 22.5·9-s − 1.02·11-s + 11.3i·13-s + 16.5·17-s + (9.54 + 16.4i)19-s + 39.4i·21-s + 35.1·23-s + 75.9i·27-s + 6.63i·29-s − 25.3i·31-s + 5.73i·33-s + 47.8i·37-s + 63.7·39-s + ⋯ |
L(s) = 1 | − 1.87i·3-s − 1.00·7-s − 2.50·9-s − 0.0928·11-s + 0.872i·13-s + 0.975·17-s + (0.502 + 0.864i)19-s + 1.88i·21-s + 1.52·23-s + 2.81i·27-s + 0.228i·29-s − 0.818i·31-s + 0.173i·33-s + 1.29i·37-s + 1.63·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.502 + 0.864i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.495669238\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.495669238\) |
\(L(2)\) |
|
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 | \( 1 + (-9.54 - 16.4i)T \) |
good | 3 | \( 1 + 5.61iT - 9T^{2} \) |
| 7 | \( 1 + 7.03T + 49T^{2} \) |
| 11 | \( 1 + 1.02T + 121T^{2} \) |
| 13 | \( 1 - 11.3iT - 169T^{2} \) |
| 17 | \( 1 - 16.5T + 289T^{2} \) |
| 23 | \( 1 - 35.1T + 529T^{2} \) |
| 29 | \( 1 - 6.63iT - 841T^{2} \) |
| 31 | \( 1 + 25.3iT - 961T^{2} \) |
| 37 | \( 1 - 47.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 12.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 50.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 78.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 45.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 9.44iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 43.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 112. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 40.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 45.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 144. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 56.8T + 6.88e3T^{2} \) |
| 89 | \( 1 - 81.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 103. iT - 9.40e3T^{2} \) |
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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
−8.764128684863991167966108822662, −7.904431369841490969427299081125, −7.25321002333140820785856561452, −6.62712455182160105789049576585, −6.01761451655819157192438745024, −5.14828362628817230650670194612, −3.53082856354622586699189429375, −2.79396354174196388648091772543, −1.68195950895413588574088757002, −0.76868292518209222117309098626,
0.56574695894402121285468394031, 2.93321006398368053205200902799, 3.18299133429248677765967766246, 4.17364551951853653755421460428, 5.19441430770949672385860530110, 5.55823705096813407237556420039, 6.66777580717340065814033963268, 7.72404484393933915468552553924, 8.751900645672196231227051989702, 9.344797959561822007517588676236