L(s) = 1 | − 2-s + 4-s + 2·5-s − 7-s − 8-s − 3·9-s − 2·10-s + 2·11-s + 3·13-s + 14-s + 16-s − 3·17-s + 3·18-s − 4·19-s + 2·20-s − 2·22-s + 23-s − 25-s − 3·26-s − 28-s + 7·31-s − 32-s + 3·34-s − 2·35-s − 3·36-s − 6·37-s + 4·38-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s − 0.353·8-s − 9-s − 0.632·10-s + 0.603·11-s + 0.832·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.707·18-s − 0.917·19-s + 0.447·20-s − 0.426·22-s + 0.208·23-s − 1/5·25-s − 0.588·26-s − 0.188·28-s + 1.25·31-s − 0.176·32-s + 0.514·34-s − 0.338·35-s − 1/2·36-s − 0.986·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270802 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8711702393\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8711702393\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 17 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.77850580443687, −12.23637942099867, −11.78882589003563, −11.28399434396645, −10.87088749538193, −10.45577408136353, −10.03049321757737, −9.413926328494634, −9.000399550201439, −8.795337999093396, −8.255955583607505, −7.791227382803644, −6.994723567141835, −6.626521536974068, −6.061578658401308, −5.991284888079881, −5.318852267095654, −4.580350100913211, −4.021362559640341, −3.425225354720739, −2.679568801364586, −2.465425973711331, −1.627024079064760, −1.234373501823467, −0.2810374179220499,
0.2810374179220499, 1.234373501823467, 1.627024079064760, 2.465425973711331, 2.679568801364586, 3.425225354720739, 4.021362559640341, 4.580350100913211, 5.318852267095654, 5.991284888079881, 6.061578658401308, 6.626521536974068, 6.994723567141835, 7.791227382803644, 8.255955583607505, 8.795337999093396, 9.000399550201439, 9.413926328494634, 10.03049321757737, 10.45577408136353, 10.87088749538193, 11.28399434396645, 11.78882589003563, 12.23637942099867, 12.77850580443687