| L(s) = 1 | − 3-s + 1.66i·5-s + 3.05·7-s + 9-s + (3.27 − 0.508i)11-s + 6.95·13-s − 1.66i·15-s + 1.66i·17-s + 4.44i·19-s − 3.05·21-s − 7.13i·23-s + 2.22·25-s − 27-s − 6.68·29-s − 3.22i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.745i·5-s + 1.15·7-s + 0.333·9-s + (0.988 − 0.153i)11-s + 1.92·13-s − 0.430i·15-s + 0.404i·17-s + 1.01i·19-s − 0.666·21-s − 1.48i·23-s + 0.444·25-s − 0.192·27-s − 1.24·29-s − 0.578i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.012688279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.012688279\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + (-3.27 + 0.508i)T \) |
| good | 5 | \( 1 - 1.66iT - 5T^{2} \) |
| 7 | \( 1 - 3.05T + 7T^{2} \) |
| 13 | \( 1 - 6.95T + 13T^{2} \) |
| 17 | \( 1 - 1.66iT - 17T^{2} \) |
| 19 | \( 1 - 4.44iT - 19T^{2} \) |
| 23 | \( 1 + 7.13iT - 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 + 3.22iT - 31T^{2} \) |
| 37 | \( 1 - 8.01iT - 37T^{2} \) |
| 41 | \( 1 + 11.2iT - 41T^{2} \) |
| 43 | \( 1 + 7.77iT - 43T^{2} \) |
| 47 | \( 1 + 8.22iT - 47T^{2} \) |
| 53 | \( 1 - 7.44iT - 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 3.88T + 67T^{2} \) |
| 71 | \( 1 + 0.887iT - 71T^{2} \) |
| 73 | \( 1 - 16.6iT - 73T^{2} \) |
| 79 | \( 1 - 5.08T + 79T^{2} \) |
| 83 | \( 1 + 13.9iT - 83T^{2} \) |
| 89 | \( 1 + 4.92T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−8.863920242536930330622344759031, −8.517723469616912942913070760236, −7.55630733602473961548635197707, −6.61880259288624593457580770544, −6.10925499181457060245853862072, −5.29221078986253950636859974821, −4.08433351058573943493449549953, −3.63589293975067188576387097605, −2.04027141809634666923161904367, −1.10962425732413203054206994285,
1.08393131918564516548090818514, 1.62462080953453113851506717190, 3.41322254066443150523944467326, 4.33482296633239212411690222460, 5.01641881037632638493810491799, 5.80610096501137048280088858302, 6.60473817994330704850558085321, 7.55394424913135203040313011665, 8.325113979504736938015771706502, 9.100886759487198029324521695834
