| L(s) = 1 | − 2·2-s + 4·4-s + 10.4·5-s − 8·8-s − 20.8·10-s − 61.1·11-s − 59.2·13-s + 16·16-s − 20.4·17-s − 80.3·19-s + 41.7·20-s + 122.·22-s + 158.·23-s − 15.8·25-s + 118.·26-s + 85.1·29-s + 243.·31-s − 32·32-s + 40.9·34-s + 290.·37-s + 160.·38-s − 83.5·40-s + 168·41-s + 7.62·43-s − 244.·44-s − 316.·46-s + 169.·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.934·5-s − 0.353·8-s − 0.660·10-s − 1.67·11-s − 1.26·13-s + 0.250·16-s − 0.291·17-s − 0.970·19-s + 0.467·20-s + 1.18·22-s + 1.43·23-s − 0.127·25-s + 0.893·26-s + 0.545·29-s + 1.41·31-s − 0.176·32-s + 0.206·34-s + 1.29·37-s + 0.685·38-s − 0.330·40-s + 0.639·41-s + 0.0270·43-s − 0.837·44-s − 1.01·46-s + 0.525·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.294001403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.294001403\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 10.4T + 125T^{2} \) |
| 11 | \( 1 + 61.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 59.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 20.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 80.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 158.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 85.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 243.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 290.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 168T + 6.89e4T^{2} \) |
| 43 | \( 1 - 7.62T + 7.95e4T^{2} \) |
| 47 | \( 1 - 169.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 250.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 805.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 33.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 277.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 631.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 768.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 418.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 761.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.57e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.04e3T + 9.12e5T^{2} \) |
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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
−9.886347350369213047179066291986, −8.985443460335952095566491266943, −8.102706811264275455903456275468, −7.31946933983935022873738461937, −6.38885327937957580439052825754, −5.43267854165545265076556557550, −4.60737212994496195560113593216, −2.69614471607526039311167843510, −2.30049847294585263309300547453, −0.66426432561399122764242441785,
0.66426432561399122764242441785, 2.30049847294585263309300547453, 2.69614471607526039311167843510, 4.60737212994496195560113593216, 5.43267854165545265076556557550, 6.38885327937957580439052825754, 7.31946933983935022873738461937, 8.102706811264275455903456275468, 8.985443460335952095566491266943, 9.886347350369213047179066291986
