| L(s) = 1 | + 5-s − 4·11-s − 13-s − 2·19-s − 8·23-s + 25-s + 2·29-s + 2·31-s − 8·37-s + 6·41-s − 10·43-s − 2·47-s − 2·53-s − 4·55-s − 8·59-s − 65-s + 4·67-s + 10·73-s − 16·79-s + 14·83-s + 2·89-s − 2·95-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.20·11-s − 0.277·13-s − 0.458·19-s − 1.66·23-s + 1/5·25-s + 0.371·29-s + 0.359·31-s − 1.31·37-s + 0.937·41-s − 1.52·43-s − 0.291·47-s − 0.274·53-s − 0.539·55-s − 1.04·59-s − 0.124·65-s + 0.488·67-s + 1.17·73-s − 1.80·79-s + 1.53·83-s + 0.211·89-s − 0.205·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3481601738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3481601738\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.94108480305559, −12.39779213465986, −12.16720291521055, −11.57850781164917, −10.92574896524671, −10.59814110988346, −10.10196804251536, −9.794584589863265, −9.291827893709755, −8.548844211455922, −8.258552491097545, −7.797371765267058, −7.303371824758408, −6.588766371407521, −6.313228855675734, −5.662730690211305, −5.201989530989414, −4.791352072992953, −4.143646757761615, −3.578503674669240, −2.896174052789551, −2.400699762797118, −1.895489669068048, −1.248415736679177, −0.1571275845840687,
0.1571275845840687, 1.248415736679177, 1.895489669068048, 2.400699762797118, 2.896174052789551, 3.578503674669240, 4.143646757761615, 4.791352072992953, 5.201989530989414, 5.662730690211305, 6.313228855675734, 6.588766371407521, 7.303371824758408, 7.797371765267058, 8.258552491097545, 8.548844211455922, 9.291827893709755, 9.794584589863265, 10.10196804251536, 10.59814110988346, 10.92574896524671, 11.57850781164917, 12.16720291521055, 12.39779213465986, 12.94108480305559
