| L(s) = 1 | − 2-s + 4-s − 1.41·5-s − 8-s + 1.41·10-s − 1.41·13-s + 16-s + 1.41·17-s − 1.41·20-s + 1.00·25-s + 1.41·26-s + 2·29-s − 32-s − 1.41·34-s + 1.41·40-s + 1.41·41-s − 1.00·50-s − 1.41·52-s − 2·58-s + 1.41·61-s + 64-s + 2.00·65-s + 1.41·68-s + 1.41·73-s − 1.41·80-s − 1.41·82-s − 2.00·85-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 1.41·5-s − 8-s + 1.41·10-s − 1.41·13-s + 16-s + 1.41·17-s − 1.41·20-s + 1.00·25-s + 1.41·26-s + 2·29-s − 32-s − 1.41·34-s + 1.41·40-s + 1.41·41-s − 1.00·50-s − 1.41·52-s − 2·58-s + 1.41·61-s + 64-s + 2.00·65-s + 1.41·68-s + 1.41·73-s − 1.41·80-s − 1.41·82-s − 2.00·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5208588155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5208588155\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.41T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.41T + T^{2} \) |
| 17 | \( 1 - 1.41T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 2T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.41T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.41T + T^{2} \) |
| 97 | \( 1 - 1.41T + T^{2} \) |
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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
−9.573851337685422654283782445352, −8.499641658719668833265027931500, −7.937278212016017953230038094861, −7.39285223842948294178162293358, −6.68520672525392673578417997868, −5.50631582440132158687708465641, −4.47532135661802338154032159259, −3.37700479978420989531265258823, −2.51730789116789760703552203771, −0.840214520965958584451282685107,
0.840214520965958584451282685107, 2.51730789116789760703552203771, 3.37700479978420989531265258823, 4.47532135661802338154032159259, 5.50631582440132158687708465641, 6.68520672525392673578417997868, 7.39285223842948294178162293358, 7.937278212016017953230038094861, 8.499641658719668833265027931500, 9.573851337685422654283782445352
