L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·6-s − 5·7-s + 9-s − 2·12-s − 6·13-s + 10·14-s − 4·16-s + 17-s − 2·18-s − 2·19-s + 5·21-s + 23-s + 12·26-s − 27-s − 10·28-s + 29-s − 5·31-s + 8·32-s − 2·34-s + 2·36-s + 7·37-s + 4·38-s + 6·39-s + 7·41-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s − 1.88·7-s + 1/3·9-s − 0.577·12-s − 1.66·13-s + 2.67·14-s − 16-s + 0.242·17-s − 0.471·18-s − 0.458·19-s + 1.09·21-s + 0.208·23-s + 2.35·26-s − 0.192·27-s − 1.88·28-s + 0.185·29-s − 0.898·31-s + 1.41·32-s − 0.342·34-s + 1/3·36-s + 1.15·37-s + 0.648·38-s + 0.960·39-s + 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.06284212792602, −12.68165504942170, −12.25288414920217, −11.92329346603056, −11.13106971259212, −10.75147340453941, −10.32412946485382, −9.814469451550098, −9.541750345402851, −9.253902604003219, −8.723526752503129, −7.980156188337239, −7.422269552730471, −7.243202899285689, −6.648326835160661, −6.200472918362076, −5.756370125496938, −4.969241692509509, −4.514459858717073, −3.830172927057493, −3.167106148656592, −2.502846871531705, −2.144977206642708, −1.150057823625039, −0.4874063002971954, 0,
0.4874063002971954, 1.150057823625039, 2.144977206642708, 2.502846871531705, 3.167106148656592, 3.830172927057493, 4.514459858717073, 4.969241692509509, 5.756370125496938, 6.200472918362076, 6.648326835160661, 7.243202899285689, 7.422269552730471, 7.980156188337239, 8.723526752503129, 9.253902604003219, 9.541750345402851, 9.814469451550098, 10.32412946485382, 10.75147340453941, 11.13106971259212, 11.92329346603056, 12.25288414920217, 12.68165504942170, 13.06284212792602