| L(s) = 1 | − 7i·3-s − 50i·7-s + 194·9-s + 121·11-s + 380i·13-s − 1.15e3i·17-s + 1.82e3·19-s − 350·21-s − 3.59e3i·23-s − 3.05e3i·27-s − 8.03e3·29-s − 2.94e3·31-s − 847i·33-s + 6.97e3i·37-s + 2.66e3·39-s + ⋯ |
| L(s) = 1 | − 0.449i·3-s − 0.385i·7-s + 0.798·9-s + 0.301·11-s + 0.623i·13-s − 0.968i·17-s + 1.15·19-s − 0.173·21-s − 1.41i·23-s − 0.807i·27-s − 1.77·29-s − 0.550·31-s − 0.135i·33-s + 0.838i·37-s + 0.280·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.115057685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.115057685\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 3 | \( 1 + 7iT - 243T^{2} \) |
| 7 | \( 1 + 50iT - 1.68e4T^{2} \) |
| 13 | \( 1 - 380iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.15e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.82e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.59e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 8.03e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.94e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.97e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 520T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.48e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 6.92e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.37e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.17e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.41e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.15e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 1.49e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.64e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 2.85e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.74e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 3.62e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.97e4iT - 8.58e9T^{2} \) |
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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
−8.971761369428194800211218926765, −7.82610667234566693887726709428, −7.15273740425347729425782677851, −6.61232539777846507559541582022, −5.43262054418980756884997467348, −4.48033221431959045235411847088, −3.62472445142636645280562348991, −2.36835243554766557255993276927, −1.35808723449671520236525505996, −0.42611698236531113692811863463,
1.08434319231107670095343860710, 2.04793364656800932560014748981, 3.47842994481973804572668659036, 3.97351512405805675860523902094, 5.32765824077635548425516347533, 5.72518767860582419641585187946, 7.08754443631387839601495574807, 7.63125761581393574153560355830, 8.708344793092519895417178738007, 9.522555311178225235240628505380
