| L(s) = 1 | − 3-s − 1.41·5-s − 1.41·7-s + 9-s + 1.41·15-s + 1.41·17-s + 1.41·19-s + 1.41·21-s + 23-s + 1.00·25-s − 27-s + 2.00·35-s − 1.41·43-s − 1.41·45-s + 1.00·49-s − 1.41·51-s + 1.41·53-s − 1.41·57-s − 1.41·63-s + 1.41·67-s − 69-s + 2·71-s − 2·73-s − 1.00·75-s + 1.41·79-s + 81-s − 2.00·85-s + ⋯ |
| L(s) = 1 | − 3-s − 1.41·5-s − 1.41·7-s + 9-s + 1.41·15-s + 1.41·17-s + 1.41·19-s + 1.41·21-s + 23-s + 1.00·25-s − 27-s + 2.00·35-s − 1.41·43-s − 1.41·45-s + 1.00·49-s − 1.41·51-s + 1.41·53-s − 1.41·57-s − 1.41·63-s + 1.41·67-s − 69-s + 2·71-s − 2·73-s − 1.00·75-s + 1.41·79-s + 81-s − 2.00·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4943860105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4943860105\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 1.41T + T^{2} \) |
| 7 | \( 1 + 1.41T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - 1.41T + T^{2} \) |
| 19 | \( 1 - 1.41T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.41T + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - 1.41T + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.41T + T^{2} \) |
| 71 | \( 1 - 2T + T^{2} \) |
| 73 | \( 1 + 2T + T^{2} \) |
| 79 | \( 1 - 1.41T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 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
−10.01282167345265033271900135351, −9.516989600066128701246766711472, −8.251601921226287768706156972987, −7.30769071981966394330638253154, −6.88105719316627692905865761815, −5.77326348534062382669245088530, −4.97077047728938472325727536269, −3.74041162880432968226639813491, −3.20078137012005517863183269263, −0.835371909794670640547955892622,
0.835371909794670640547955892622, 3.20078137012005517863183269263, 3.74041162880432968226639813491, 4.97077047728938472325727536269, 5.77326348534062382669245088530, 6.88105719316627692905865761815, 7.30769071981966394330638253154, 8.251601921226287768706156972987, 9.516989600066128701246766711472, 10.01282167345265033271900135351
