| L(s) = 1 | + 2·2-s + 2·4-s + 3·7-s − 2·11-s − 13-s + 6·14-s − 4·16-s + 2·17-s − 5·19-s − 4·22-s + 6·23-s − 2·26-s + 6·28-s − 10·29-s − 3·31-s − 8·32-s + 4·34-s − 2·37-s − 10·38-s + 8·41-s − 43-s − 4·44-s + 12·46-s + 2·47-s + 2·49-s − 2·52-s − 4·53-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.13·7-s − 0.603·11-s − 0.277·13-s + 1.60·14-s − 16-s + 0.485·17-s − 1.14·19-s − 0.852·22-s + 1.25·23-s − 0.392·26-s + 1.13·28-s − 1.85·29-s − 0.538·31-s − 1.41·32-s + 0.685·34-s − 0.328·37-s − 1.62·38-s + 1.24·41-s − 0.152·43-s − 0.603·44-s + 1.76·46-s + 0.291·47-s + 2/7·49-s − 0.277·52-s − 0.549·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.411336342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.411336342\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−12.58033395895800211021523424017, −11.37636969058177948973372420565, −10.84373752796776504705892968641, −9.294135845295806835405725735271, −8.102126663118456650752904190012, −6.99182605075727565742724929802, −5.60694328736998842121660037613, −4.92213034669592094010306499374, −3.78545707329788062312381068333, −2.27079729708413704097834802658,
2.27079729708413704097834802658, 3.78545707329788062312381068333, 4.92213034669592094010306499374, 5.60694328736998842121660037613, 6.99182605075727565742724929802, 8.102126663118456650752904190012, 9.294135845295806835405725735271, 10.84373752796776504705892968641, 11.37636969058177948973372420565, 12.58033395895800211021523424017
