| L(s) = 1 | − 2·2-s + 2·4-s − 4·5-s − 7-s + 8·10-s − 11-s + 2·14-s − 4·16-s + 2·17-s + 5·19-s − 8·20-s + 2·22-s − 3·23-s + 11·25-s − 2·28-s + 2·31-s + 8·32-s − 4·34-s + 4·35-s + 9·37-s − 10·38-s − 3·41-s − 2·44-s + 6·46-s − 10·47-s + 49-s − 22·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.78·5-s − 0.377·7-s + 2.52·10-s − 0.301·11-s + 0.534·14-s − 16-s + 0.485·17-s + 1.14·19-s − 1.78·20-s + 0.426·22-s − 0.625·23-s + 11/5·25-s − 0.377·28-s + 0.359·31-s + 1.41·32-s − 0.685·34-s + 0.676·35-s + 1.47·37-s − 1.62·38-s − 0.468·41-s − 0.301·44-s + 0.884·46-s − 1.45·47-s + 1/7·49-s − 3.11·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3536171919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3536171919\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−13.53476650906437, −13.00249590518438, −12.47092973177182, −11.84143027036759, −11.55535801654527, −11.24352361345642, −10.57267316841913, −10.02938230821188, −9.775528413749125, −9.105060101029762, −8.561997103297477, −8.144477089211587, −7.791422009576266, −7.301992866730267, −7.022188594182071, −6.255378127687907, −5.604170821978770, −4.746559948652641, −4.407793697477342, −3.702721474112352, −3.124329890390158, −2.637006414680820, −1.590064170188083, −0.9435614620642101, −0.2985010244481466,
0.2985010244481466, 0.9435614620642101, 1.590064170188083, 2.637006414680820, 3.124329890390158, 3.702721474112352, 4.407793697477342, 4.746559948652641, 5.604170821978770, 6.255378127687907, 7.022188594182071, 7.301992866730267, 7.791422009576266, 8.144477089211587, 8.561997103297477, 9.105060101029762, 9.775528413749125, 10.02938230821188, 10.57267316841913, 11.24352361345642, 11.55535801654527, 11.84143027036759, 12.47092973177182, 13.00249590518438, 13.53476650906437
