| L(s) = 1 | − 22·2-s − 28·4-s + 625·5-s − 5.98e3·7-s + 1.18e4·8-s − 1.37e4·10-s + 1.46e4·11-s + 3.79e4·13-s + 1.31e5·14-s − 2.47e5·16-s + 4.41e5·17-s + 4.41e5·19-s − 1.75e4·20-s − 3.22e5·22-s − 2.26e6·23-s + 3.90e5·25-s − 8.33e5·26-s + 1.67e5·28-s + 1.04e6·29-s − 7.91e6·31-s − 6.48e5·32-s − 9.70e6·34-s − 3.74e6·35-s − 2.09e7·37-s − 9.72e6·38-s + 7.42e6·40-s − 1.32e7·41-s + ⋯ |
| L(s) = 1 | − 0.972·2-s − 0.0546·4-s + 0.447·5-s − 0.942·7-s + 1.02·8-s − 0.434·10-s + 0.301·11-s + 0.368·13-s + 0.916·14-s − 0.942·16-s + 1.28·17-s + 0.777·19-s − 0.0244·20-s − 0.293·22-s − 1.68·23-s + 1/5·25-s − 0.357·26-s + 0.0515·28-s + 0.275·29-s − 1.53·31-s − 0.109·32-s − 1.24·34-s − 0.421·35-s − 1.84·37-s − 0.756·38-s + 0.458·40-s − 0.734·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - p^{4} T \) |
| good | 2 | \( 1 + 11 p T + p^{9} T^{2} \) |
| 7 | \( 1 + 5988 T + p^{9} T^{2} \) |
| 11 | \( 1 - 14648 T + p^{9} T^{2} \) |
| 13 | \( 1 - 37906 T + p^{9} T^{2} \) |
| 17 | \( 1 - 441098 T + p^{9} T^{2} \) |
| 19 | \( 1 - 441820 T + p^{9} T^{2} \) |
| 23 | \( 1 + 2264136 T + p^{9} T^{2} \) |
| 29 | \( 1 - 1049350 T + p^{9} T^{2} \) |
| 31 | \( 1 + 7910568 T + p^{9} T^{2} \) |
| 37 | \( 1 + 20992558 T + p^{9} T^{2} \) |
| 41 | \( 1 + 13285562 T + p^{9} T^{2} \) |
| 43 | \( 1 + 23130764 T + p^{9} T^{2} \) |
| 47 | \( 1 - 13873688 T + p^{9} T^{2} \) |
| 53 | \( 1 - 57635174 T + p^{9} T^{2} \) |
| 59 | \( 1 - 32042120 T + p^{9} T^{2} \) |
| 61 | \( 1 - 110664022 T + p^{9} T^{2} \) |
| 67 | \( 1 + 118568268 T + p^{9} T^{2} \) |
| 71 | \( 1 + 276679712 T + p^{9} T^{2} \) |
| 73 | \( 1 + 264023294 T + p^{9} T^{2} \) |
| 79 | \( 1 - 448202760 T + p^{9} T^{2} \) |
| 83 | \( 1 + 851015796 T + p^{9} T^{2} \) |
| 89 | \( 1 + 189894930 T + p^{9} T^{2} \) |
| 97 | \( 1 + 1014149278 T + p^{9} T^{2} \) |
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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
−13.35908158315603179776196003157, −12.03448456716295217626826906609, −10.31249255888262332910500692750, −9.681587704053494840038697779293, −8.523805674521537050713662808047, −7.13989808581356776603040401704, −5.58990964215608220534294025213, −3.59908098270450039348688894149, −1.52577136364029318023067120025, 0,
1.52577136364029318023067120025, 3.59908098270450039348688894149, 5.58990964215608220534294025213, 7.13989808581356776603040401704, 8.523805674521537050713662808047, 9.681587704053494840038697779293, 10.31249255888262332910500692750, 12.03448456716295217626826906609, 13.35908158315603179776196003157
