| L(s) = 1 | + 2·3-s − 7-s + 9-s + 6·17-s + 6·19-s − 2·21-s − 5·25-s − 4·27-s + 6·29-s + 8·31-s + 2·37-s − 2·41-s − 8·43-s − 8·47-s + 49-s + 12·51-s − 2·53-s + 12·57-s + 6·59-s − 63-s − 12·67-s − 8·71-s − 6·73-s − 10·75-s + 16·79-s − 11·81-s + 6·83-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.45·17-s + 1.37·19-s − 0.436·21-s − 25-s − 0.769·27-s + 1.11·29-s + 1.43·31-s + 0.328·37-s − 0.312·41-s − 1.21·43-s − 1.16·47-s + 1/7·49-s + 1.68·51-s − 0.274·53-s + 1.58·57-s + 0.781·59-s − 0.125·63-s − 1.46·67-s − 0.949·71-s − 0.702·73-s − 1.15·75-s + 1.80·79-s − 1.22·81-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.258187311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.258187311\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.24785251925223, −12.32049733038955, −11.97004971174728, −11.71161024956699, −11.13186148732095, −10.24371093081516, −9.942187548021264, −9.787572939851736, −9.182734885545141, −8.545812298058480, −8.298332120963735, −7.655873111419865, −7.523439693292140, −6.807313956329382, −6.072034536859189, −5.912700654239842, −4.969705474315885, −4.802665458525208, −3.814123525336070, −3.471454625192857, −3.001839281110566, −2.653649680565848, −1.818108578211511, −1.259038105022751, −0.5364673163082127,
0.5364673163082127, 1.259038105022751, 1.818108578211511, 2.653649680565848, 3.001839281110566, 3.471454625192857, 3.814123525336070, 4.802665458525208, 4.969705474315885, 5.912700654239842, 6.072034536859189, 6.807313956329382, 7.523439693292140, 7.655873111419865, 8.298332120963735, 8.545812298058480, 9.182734885545141, 9.787572939851736, 9.942187548021264, 10.24371093081516, 11.13186148732095, 11.71161024956699, 11.97004971174728, 12.32049733038955, 13.24785251925223
