| L(s) = 1 | − 3-s + 4-s + 1.41·5-s + 9-s − 12-s − 1.41·15-s + 16-s + 1.41·20-s + 1.00·25-s − 27-s − 1.41·29-s + 36-s − 1.41·41-s + 1.41·45-s − 1.41·47-s − 48-s + 49-s − 1.41·60-s + 64-s − 1.00·75-s + 1.41·80-s + 81-s − 1.41·83-s + 1.41·87-s + 1.41·89-s + 1.00·100-s + 1.41·101-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 1.41·5-s + 9-s − 12-s − 1.41·15-s + 16-s + 1.41·20-s + 1.00·25-s − 27-s − 1.41·29-s + 36-s − 1.41·41-s + 1.41·45-s − 1.41·47-s − 48-s + 49-s − 1.41·60-s + 64-s − 1.00·75-s + 1.41·80-s + 81-s − 1.41·83-s + 1.41·87-s + 1.41·89-s + 1.00·100-s + 1.41·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.351175719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.351175719\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 577 | \( 1 + T \) |
| good | 2 | \( 1 - T^{2} \) |
| 5 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.41T + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 1.41T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.41T + T^{2} \) |
| 89 | \( 1 - 1.41T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.905908526461293479628511552318, −8.878647618936286433481458572954, −7.64143248660170037488745310983, −6.89454614095396951566066104136, −6.20630894115600210121085951125, −5.66446373524480503516355982133, −4.92938000384185026744593529881, −3.52722472906205134012410944167, −2.21474812847995134928188326086, −1.47740465680117519430118401586,
1.47740465680117519430118401586, 2.21474812847995134928188326086, 3.52722472906205134012410944167, 4.92938000384185026744593529881, 5.66446373524480503516355982133, 6.20630894115600210121085951125, 6.89454614095396951566066104136, 7.64143248660170037488745310983, 8.878647618936286433481458572954, 9.905908526461293479628511552318
