L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s − 25·5-s + 36·6-s + 11.1·7-s − 64·8-s + 81·9-s + 100·10-s − 345.·11-s − 144·12-s + 862.·13-s − 44.6·14-s + 225·15-s + 256·16-s − 1.13e3·17-s − 324·18-s − 361·19-s − 400·20-s − 100.·21-s + 1.38e3·22-s − 1.60e3·23-s + 576·24-s + 625·25-s − 3.44e3·26-s − 729·27-s + 178.·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 + 0.0861·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.859·11-s − 0.288·12-s + 1.41·13-s − 0.0608·14-s + 0.258·15-s + 0.250·16-s − 0.952·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s − 0.0497·21-s + 0.607·22-s − 0.633·23-s + 0.204·24-s + 0.200·25-s − 1.00·26-s − 0.192·27-s + 0.0430·28-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
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 - 11.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + 345.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 862.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.13e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.60e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.56e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.97e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.63e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.11e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.48e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.66e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.41e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.91e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.82e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.43e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.07e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.24e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.12e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.94e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.36e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.09e4T + 8.58e9T^{2} \) |
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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
−9.615752846542737058959200070924, −8.401100598211052135849023385750, −8.020122815655680643573143624494, −6.74663865089821004815682438113, −6.10361863536025115934874463557, −4.88854500166490961733725211834, −3.77371376264956206358122678913, −2.40167061188022767120274126164, −1.06500716950828249459314824598, 0,
1.06500716950828249459314824598, 2.40167061188022767120274126164, 3.77371376264956206358122678913, 4.88854500166490961733725211834, 6.10361863536025115934874463557, 6.74663865089821004815682438113, 8.020122815655680643573143624494, 8.401100598211052135849023385750, 9.615752846542737058959200070924