| L(s) = 1 | − 2-s + 3·3-s − 7·4-s + 5·5-s − 3·6-s + 4·7-s + 15·8-s + 9·9-s − 5·10-s − 36·11-s − 21·12-s − 22·13-s − 4·14-s + 15·15-s + 41·16-s − 2·17-s − 9·18-s − 56·19-s − 35·20-s + 12·21-s + 36·22-s − 40·23-s + 45·24-s + 25·25-s + 22·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 0.577·3-s − 7/8·4-s + 0.447·5-s − 0.204·6-s + 0.215·7-s + 0.662·8-s + 1/3·9-s − 0.158·10-s − 0.986·11-s − 0.505·12-s − 0.469·13-s − 0.0763·14-s + 0.258·15-s + 0.640·16-s − 0.0285·17-s − 0.117·18-s − 0.676·19-s − 0.391·20-s + 0.124·21-s + 0.348·22-s − 0.362·23-s + 0.382·24-s + 1/5·25-s + 0.165·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 29 | \( 1 - p T \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 17 | \( 1 + 2 T + p^{3} T^{2} \) |
| 19 | \( 1 + 56 T + p^{3} T^{2} \) |
| 23 | \( 1 + 40 T + p^{3} T^{2} \) |
| 31 | \( 1 - 152 T + p^{3} T^{2} \) |
| 37 | \( 1 - 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 250 T + p^{3} T^{2} \) |
| 43 | \( 1 + 412 T + p^{3} T^{2} \) |
| 47 | \( 1 + 120 T + p^{3} T^{2} \) |
| 53 | \( 1 + 762 T + p^{3} T^{2} \) |
| 59 | \( 1 + 188 T + p^{3} T^{2} \) |
| 61 | \( 1 + 54 T + p^{3} T^{2} \) |
| 67 | \( 1 + 244 T + p^{3} T^{2} \) |
| 71 | \( 1 - 600 T + p^{3} T^{2} \) |
| 73 | \( 1 - 6 T + p^{3} T^{2} \) |
| 79 | \( 1 + 640 T + p^{3} T^{2} \) |
| 83 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 89 | \( 1 - 150 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1690 T + p^{3} T^{2} \) |
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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
−10.03337444848525061151970572573, −9.453757669856857417113204759096, −8.323091959849150548058122240642, −7.959638156522876811417947654592, −6.62038039851849601214079969333, −5.24153362785041470946393272646, −4.47289470116913976637026712426, −3.05582687556138009517029285414, −1.71359044527595527357752911774, 0,
1.71359044527595527357752911774, 3.05582687556138009517029285414, 4.47289470116913976637026712426, 5.24153362785041470946393272646, 6.62038039851849601214079969333, 7.959638156522876811417947654592, 8.323091959849150548058122240642, 9.453757669856857417113204759096, 10.03337444848525061151970572573
