| L(s) = 1 | − 2-s + 2·3-s − 4-s + 5-s − 2·6-s + 2·7-s + 3·8-s + 9-s − 10-s − 2·12-s − 13-s − 2·14-s + 2·15-s − 16-s + 5·17-s − 18-s − 6·19-s − 20-s + 4·21-s + 2·23-s + 6·24-s − 4·25-s + 26-s − 4·27-s − 2·28-s − 9·29-s − 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 1/2·4-s + 0.447·5-s − 0.816·6-s + 0.755·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.577·12-s − 0.277·13-s − 0.534·14-s + 0.516·15-s − 1/4·16-s + 1.21·17-s − 0.235·18-s − 1.37·19-s − 0.223·20-s + 0.872·21-s + 0.417·23-s + 1.22·24-s − 4/5·25-s + 0.196·26-s − 0.769·27-s − 0.377·28-s − 1.67·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.019794861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019794861\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.66555884841707973986817495177, −12.70933619053980848016905989778, −11.10272305542928838383025084052, −9.950218896470013175355990146225, −9.127942331541421284163370204249, −8.264856530445176716222542589828, −7.49323810887805876503518242868, −5.44861874763839175210229504778, −3.89751875979123562820900427499, −1.98218366620434510274683932709,
1.98218366620434510274683932709, 3.89751875979123562820900427499, 5.44861874763839175210229504778, 7.49323810887805876503518242868, 8.264856530445176716222542589828, 9.127942331541421284163370204249, 9.950218896470013175355990146225, 11.10272305542928838383025084052, 12.70933619053980848016905989778, 13.66555884841707973986817495177
