| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 3·8-s + 9-s + 2·11-s − 12-s − 13-s − 16-s + 6·17-s − 18-s − 4·19-s − 2·22-s − 4·23-s + 3·24-s + 26-s + 27-s − 6·29-s + 4·31-s − 5·32-s + 2·33-s − 6·34-s − 36-s + 10·37-s + 4·38-s − 39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s − 0.277·13-s − 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.426·22-s − 0.834·23-s + 0.612·24-s + 0.196·26-s + 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.883·32-s + 0.348·33-s − 1.02·34-s − 1/6·36-s + 1.64·37-s + 0.648·38-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.521118075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.521118075\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.54708698621842, −14.25433576501511, −13.46627636829116, −13.20957282039406, −12.43419687408147, −12.19488796080709, −11.32620679053252, −10.85311074143782, −10.18894886747978, −9.731324286031133, −9.366528150627450, −8.950236811592950, −8.052286883905937, −7.929059728568659, −7.526199880455519, −6.533942834153056, −6.151319341742592, −5.307261657917124, −4.659596123649121, −4.095103117598451, −3.574644194241308, −2.812370544657431, −1.924726487691035, −1.359677850478721, −0.5054364258342413,
0.5054364258342413, 1.359677850478721, 1.924726487691035, 2.812370544657431, 3.574644194241308, 4.095103117598451, 4.659596123649121, 5.307261657917124, 6.151319341742592, 6.533942834153056, 7.526199880455519, 7.929059728568659, 8.052286883905937, 8.950236811592950, 9.366528150627450, 9.731324286031133, 10.18894886747978, 10.85311074143782, 11.32620679053252, 12.19488796080709, 12.43419687408147, 13.20957282039406, 13.46627636829116, 14.25433576501511, 14.54708698621842
