| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s − 16·5-s + 36·6-s − 49·7-s + 64·8-s + 81·9-s − 64·10-s − 114·11-s + 144·12-s − 169·13-s − 196·14-s − 144·15-s + 256·16-s + 538·17-s + 324·18-s − 536·19-s − 256·20-s − 441·21-s − 456·22-s − 4.59e3·23-s + 576·24-s − 2.86e3·25-s − 676·26-s + 729·27-s − 784·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.286·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.202·10-s − 0.284·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.165·15-s + 1/4·16-s + 0.451·17-s + 0.235·18-s − 0.340·19-s − 0.143·20-s − 0.218·21-s − 0.200·22-s − 1.81·23-s + 0.204·24-s − 0.918·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{2} T \) |
| 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| 13 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 + 16 T + p^{5} T^{2} \) |
| 11 | \( 1 + 114 T + p^{5} T^{2} \) |
| 17 | \( 1 - 538 T + p^{5} T^{2} \) |
| 19 | \( 1 + 536 T + p^{5} T^{2} \) |
| 23 | \( 1 + 4596 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1594 T + p^{5} T^{2} \) |
| 31 | \( 1 - 9364 T + p^{5} T^{2} \) |
| 37 | \( 1 + 12002 T + p^{5} T^{2} \) |
| 41 | \( 1 - 4928 T + p^{5} T^{2} \) |
| 43 | \( 1 + 14284 T + p^{5} T^{2} \) |
| 47 | \( 1 + 22262 T + p^{5} T^{2} \) |
| 53 | \( 1 + 474 T + p^{5} T^{2} \) |
| 59 | \( 1 + 4182 T + p^{5} T^{2} \) |
| 61 | \( 1 + 21830 T + p^{5} T^{2} \) |
| 67 | \( 1 - 20780 T + p^{5} T^{2} \) |
| 71 | \( 1 - 18682 T + p^{5} T^{2} \) |
| 73 | \( 1 + 37866 T + p^{5} T^{2} \) |
| 79 | \( 1 + 27840 T + p^{5} T^{2} \) |
| 83 | \( 1 + 101914 T + p^{5} T^{2} \) |
| 89 | \( 1 + 77644 T + p^{5} T^{2} \) |
| 97 | \( 1 + 60050 T + p^{5} T^{2} \) |
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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
−9.851754364247989144447430386776, −8.435819324377670253822432922997, −7.83822127719836991178545218090, −6.76836560156937313579698943830, −5.87745029768817222911068808661, −4.68533717589679297265900060254, −3.77054644278128225523809492652, −2.83814754167294671741018492050, −1.72601641530235749163359466802, 0,
1.72601641530235749163359466802, 2.83814754167294671741018492050, 3.77054644278128225523809492652, 4.68533717589679297265900060254, 5.87745029768817222911068808661, 6.76836560156937313579698943830, 7.83822127719836991178545218090, 8.435819324377670253822432922997, 9.851754364247989144447430386776
