| L(s) = 1 | − 11-s + 2·13-s + 2·17-s + 4·19-s + 2·29-s + 2·37-s − 2·41-s − 12·43-s − 8·47-s − 7·49-s + 6·53-s − 12·59-s + 6·61-s + 4·67-s + 6·73-s + 16·79-s − 4·83-s − 10·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.301·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 0.371·29-s + 0.328·37-s − 0.312·41-s − 1.82·43-s − 1.16·47-s − 49-s + 0.824·53-s − 1.56·59-s + 0.768·61-s + 0.488·67-s + 0.702·73-s + 1.80·79-s − 0.439·83-s − 1.05·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.10424801539830, −14.53212330152108, −13.97714075961885, −13.45275641786182, −13.14240272621707, −12.31935425165947, −12.04755947769102, −11.24106256333435, −11.07130935031487, −10.16447283300850, −9.865339668715721, −9.315267117109881, −8.568312024655072, −8.090190127993308, −7.684551090925712, −6.826276174083222, −6.496916512454087, −5.715166642950943, −5.152530126703934, −4.692617396296915, −3.741591654938592, −3.314435250312068, −2.634034487166566, −1.699567713759099, −1.069608895575871, 0,
1.069608895575871, 1.699567713759099, 2.634034487166566, 3.314435250312068, 3.741591654938592, 4.692617396296915, 5.152530126703934, 5.715166642950943, 6.496916512454087, 6.826276174083222, 7.684551090925712, 8.090190127993308, 8.568312024655072, 9.315267117109881, 9.865339668715721, 10.16447283300850, 11.07130935031487, 11.24106256333435, 12.04755947769102, 12.31935425165947, 13.14240272621707, 13.45275641786182, 13.97714075961885, 14.53212330152108, 15.10424801539830
