L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s − 25·5-s + 36·6-s + 135.·7-s + 64·8-s + 81·9-s − 100·10-s + 128.·11-s + 144·12-s + 150.·13-s + 541.·14-s − 225·15-s + 256·16-s + 1.04e3·17-s + 324·18-s + 361·19-s − 400·20-s + 1.21e3·21-s + 512.·22-s − 2.73e3·23-s + 576·24-s + 625·25-s + 601.·26-s + 729·27-s + 2.16e3·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 1.04·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.319·11-s + 0.288·12-s + 0.246·13-s + 0.738·14-s − 0.258·15-s + 0.250·16-s + 0.875·17-s + 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.602·21-s + 0.225·22-s − 1.07·23-s + 0.204·24-s + 0.200·25-s + 0.174·26-s + 0.192·27-s + 0.521·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(5.170522792\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.170522792\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 - 9T \) |
| 5 | \( 1 + 25T \) |
| 19 | \( 1 - 361T \) |
good | 7 | \( 1 - 135.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 128.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 150.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.04e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 2.73e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 726.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.86e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.59e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.48e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.98e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.67e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.81e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.35e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.36e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.99e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.74e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.46e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.87e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.06e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.17e5T + 8.58e9T^{2} \) |
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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
−10.10732896059453788526266885717, −8.895873306042484885326643067306, −8.022823965879490406441459765607, −7.44122442542900098456978846791, −6.24735538699608588948342360748, −5.15087464067281144126920279356, −4.23234939128357531933870208748, −3.38745018509729449192339452054, −2.15274721656084113114767685343, −1.04702231396796463996467243018,
1.04702231396796463996467243018, 2.15274721656084113114767685343, 3.38745018509729449192339452054, 4.23234939128357531933870208748, 5.15087464067281144126920279356, 6.24735538699608588948342360748, 7.44122442542900098456978846791, 8.022823965879490406441459765607, 8.895873306042484885326643067306, 10.10732896059453788526266885717