L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 4·11-s + 2·13-s + 15-s + 2·17-s + 4·19-s − 21-s − 23-s + 25-s − 27-s + 2·29-s + 8·31-s − 4·33-s − 35-s − 6·37-s − 2·39-s − 6·41-s + 4·43-s − 45-s + 8·47-s + 49-s − 2·51-s + 10·53-s − 4·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.258·15-s + 0.485·17-s + 0.917·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.696·33-s − 0.169·35-s − 0.986·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s + 1.37·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.128220783\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.128220783\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.55029038801771, −12.58024561081712, −12.19176676222592, −11.93091946306017, −11.47744120468210, −11.07456034621376, −10.45756436169762, −10.02470620480283, −9.565991422946373, −8.834900007618631, −8.574928323261466, −7.968293594145623, −7.387883763687463, −6.946562459801479, −6.430792329117344, −5.957338382666181, −5.314416386506627, −4.922346514469451, −4.189851065403936, −3.813547944072830, −3.295808371066513, −2.500502210238715, −1.702443645138217, −1.015085230686127, −0.6769107776514370,
0.6769107776514370, 1.015085230686127, 1.702443645138217, 2.500502210238715, 3.295808371066513, 3.813547944072830, 4.189851065403936, 4.922346514469451, 5.314416386506627, 5.957338382666181, 6.430792329117344, 6.946562459801479, 7.387883763687463, 7.968293594145623, 8.574928323261466, 8.834900007618631, 9.565991422946373, 10.02470620480283, 10.45756436169762, 11.07456034621376, 11.47744120468210, 11.93091946306017, 12.19176676222592, 12.58024561081712, 13.55029038801771