| L(s) = 1 | − 2·2-s + 2·4-s + 5-s − 2·7-s − 2·10-s + 11-s − 13-s + 4·14-s − 4·16-s − 5·19-s + 2·20-s − 2·22-s + 8·23-s + 25-s + 2·26-s − 4·28-s − 9·29-s − 4·31-s + 8·32-s − 2·35-s + 2·37-s + 10·38-s + 5·41-s − 12·43-s + 2·44-s − 16·46-s − 12·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s − 0.755·7-s − 0.632·10-s + 0.301·11-s − 0.277·13-s + 1.06·14-s − 16-s − 1.14·19-s + 0.447·20-s − 0.426·22-s + 1.66·23-s + 1/5·25-s + 0.392·26-s − 0.755·28-s − 1.67·29-s − 0.718·31-s + 1.41·32-s − 0.338·35-s + 0.328·37-s + 1.62·38-s + 0.780·41-s − 1.82·43-s + 0.301·44-s − 2.35·46-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.48628526882767, −13.15900171635799, −12.75042317650761, −12.43116610027340, −11.35889108858298, −11.20765803987492, −10.88905970087655, −10.14249311275966, −9.692919274750289, −9.540039930306232, −8.950026163091814, −8.605636686400737, −8.031786942884509, −7.455345336480198, −6.998483451298117, −6.536909343906740, −6.163950712291533, −5.397293193316767, −4.807728297073160, −4.308927702011832, −3.407467041643500, −3.055136257034897, −2.204032198833574, −1.692862834003435, −1.182579774279163, 0, 0,
1.182579774279163, 1.692862834003435, 2.204032198833574, 3.055136257034897, 3.407467041643500, 4.308927702011832, 4.807728297073160, 5.397293193316767, 6.163950712291533, 6.536909343906740, 6.998483451298117, 7.455345336480198, 8.031786942884509, 8.605636686400737, 8.950026163091814, 9.540039930306232, 9.692919274750289, 10.14249311275966, 10.88905970087655, 11.20765803987492, 11.35889108858298, 12.43116610027340, 12.75042317650761, 13.15900171635799, 13.48628526882767
