| L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s − 7-s + 3·8-s + 9-s + 10-s + 12-s − 6·13-s + 14-s + 15-s − 16-s + 2·17-s − 18-s + 8·19-s + 20-s + 21-s + 8·23-s − 3·24-s + 25-s + 6·26-s − 27-s + 28-s + 2·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 1.66·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s + 1.83·19-s + 0.223·20-s + 0.218·21-s + 1.66·23-s − 0.612·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s + 0.188·28-s + 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.159885632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.159885632\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 43 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.03433496415689, −12.45495411807232, −12.13024497052551, −11.75165297589228, −11.15926791802938, −10.65543652112337, −10.08109334147827, −9.842361973940962, −9.354104250560877, −8.997135859127688, −8.289928915652959, −7.753181195757115, −7.458316262223869, −6.851403209376034, −6.676315405702107, −5.419646021625341, −5.308617829889365, −4.959061551456270, −4.192445145561699, −3.748245190526050, −2.920436545252566, −2.577572102925725, −1.501738194030141, −0.8158706325588545, −0.5322097624283987,
0.5322097624283987, 0.8158706325588545, 1.501738194030141, 2.577572102925725, 2.920436545252566, 3.748245190526050, 4.192445145561699, 4.959061551456270, 5.308617829889365, 5.419646021625341, 6.676315405702107, 6.851403209376034, 7.458316262223869, 7.753181195757115, 8.289928915652959, 8.997135859127688, 9.354104250560877, 9.842361973940962, 10.08109334147827, 10.65543652112337, 11.15926791802938, 11.75165297589228, 12.13024497052551, 12.45495411807232, 13.03433496415689
