| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s + 2·7-s − 3·8-s + 9-s − 11-s + 12-s + 2·14-s − 16-s + 18-s − 2·19-s − 2·21-s − 22-s + 23-s + 3·24-s − 5·25-s − 27-s − 2·28-s − 10·29-s − 4·31-s + 5·32-s + 33-s − 36-s − 2·37-s − 2·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.755·7-s − 1.06·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 0.534·14-s − 1/4·16-s + 0.235·18-s − 0.458·19-s − 0.436·21-s − 0.213·22-s + 0.208·23-s + 0.612·24-s − 25-s − 0.192·27-s − 0.377·28-s − 1.85·29-s − 0.718·31-s + 0.883·32-s + 0.174·33-s − 1/6·36-s − 0.328·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128271 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128271 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8933896582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8933896582\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−13.41026305720696, −13.05158085169672, −12.59510321947257, −12.14429295284617, −11.62041558236618, −11.14114383828775, −10.81745235586309, −10.16417188592095, −9.545580024667254, −9.209072916087548, −8.606779255485796, −8.064984120871747, −7.521996597555198, −7.091946393983775, −6.257783077424973, −5.845168544812811, −5.396778860569447, −4.973711484313525, −4.416651995583703, −3.821203865401426, −3.536209223097084, −2.524709355132623, −1.990730208916066, −1.221894239416051, −0.2740697612663297,
0.2740697612663297, 1.221894239416051, 1.990730208916066, 2.524709355132623, 3.536209223097084, 3.821203865401426, 4.416651995583703, 4.973711484313525, 5.396778860569447, 5.845168544812811, 6.257783077424973, 7.091946393983775, 7.521996597555198, 8.064984120871747, 8.606779255485796, 9.209072916087548, 9.545580024667254, 10.16417188592095, 10.81745235586309, 11.14114383828775, 11.62041558236618, 12.14429295284617, 12.59510321947257, 13.05158085169672, 13.41026305720696
