| L(s) = 1 | − 4·7-s + 2·19-s + 8·25-s + 20·29-s − 20·41-s − 24·43-s − 2·49-s − 20·53-s − 24·59-s + 16·61-s − 16·71-s + 12·73-s − 12·89-s − 16·107-s − 12·113-s + 4·121-s + 127-s + 131-s − 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 18·169-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.458·19-s + 8/5·25-s + 3.71·29-s − 3.12·41-s − 3.65·43-s − 2/7·49-s − 2.74·53-s − 3.12·59-s + 2.04·61-s − 1.89·71-s + 1.40·73-s − 1.27·89-s − 1.54·107-s − 1.12·113-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s − 0.693·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.38·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7014118220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7014118220\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 148 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.347453729947505209716856214190, −8.414066668104710553780849846131, −8.388607560446760135603325460951, −8.097423073832669930039587754061, −7.47295660411565597160686734721, −6.71360108475616984799355836452, −6.68879116065859589580559759584, −6.51021849318124771537312908184, −6.32466854167710440351249863080, −5.36623782528449259751731534900, −5.11515380299465485510014742758, −4.65057334902231568529829007215, −4.55508378514584626340535146117, −3.44589469951260031632651024228, −3.35727701118365112113449531692, −2.97076155210468261316761445450, −2.71705027548671942655146982522, −1.53610189667986003881240686396, −1.41449687756811312408534331126, −0.27512453030734254081306282151,
0.27512453030734254081306282151, 1.41449687756811312408534331126, 1.53610189667986003881240686396, 2.71705027548671942655146982522, 2.97076155210468261316761445450, 3.35727701118365112113449531692, 3.44589469951260031632651024228, 4.55508378514584626340535146117, 4.65057334902231568529829007215, 5.11515380299465485510014742758, 5.36623782528449259751731534900, 6.32466854167710440351249863080, 6.51021849318124771537312908184, 6.68879116065859589580559759584, 6.71360108475616984799355836452, 7.47295660411565597160686734721, 8.097423073832669930039587754061, 8.388607560446760135603325460951, 8.414066668104710553780849846131, 9.347453729947505209716856214190
