| L(s) = 1 | − i·3-s + (2 − i)5-s − 2i·7-s − 9-s − 2·11-s + 2i·13-s + (−1 − 2i)15-s − 6i·17-s + 8·19-s − 2·21-s + 4i·23-s + (3 − 4i)25-s + i·27-s − 8·29-s + 2i·33-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (0.894 − 0.447i)5-s − 0.755i·7-s − 0.333·9-s − 0.603·11-s + 0.554i·13-s + (−0.258 − 0.516i)15-s − 1.45i·17-s + 1.83·19-s − 0.436·21-s + 0.834i·23-s + (0.600 − 0.800i)25-s + 0.192i·27-s − 1.48·29-s + 0.348i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16037 - 0.717152i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16037 - 0.717152i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-2 + i)T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−11.95747170491625376936456688781, −11.14301650004479666707636163591, −9.794185847406901684526661674217, −9.290581578561918871706111188724, −7.78894294295189177838716116071, −7.08380431398268421522064520358, −5.76860183667155501762310937818, −4.80823266589401103720683683811, −2.96295509022404813023262359067, −1.29364908428724414421228242225,
2.26070678660169631569448024988, 3.56253821189029977220497335255, 5.37928983360670206917545594577, 5.81862689959597119626830896643, 7.33059866362959324611073021213, 8.586148689580962688118496469408, 9.517796251947310452634747002712, 10.35851928869529609632388317452, 11.09787388763999268452258558317, 12.36670383016634175464617955309
