| L(s) = 1 | + 4·4-s + 6·11-s + 12·16-s + 2·19-s − 18·29-s − 2·31-s − 6·41-s + 24·44-s + 10·49-s + 6·59-s − 20·61-s + 32·64-s + 6·71-s + 8·76-s + 32·79-s + 30·89-s − 18·101-s + 2·109-s − 72·116-s + 5·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2·4-s + 1.80·11-s + 3·16-s + 0.458·19-s − 3.34·29-s − 0.359·31-s − 0.937·41-s + 3.61·44-s + 10/7·49-s + 0.781·59-s − 2.56·61-s + 4·64-s + 0.712·71-s + 0.917·76-s + 3.60·79-s + 3.17·89-s − 1.79·101-s + 0.191·109-s − 6.68·116-s + 5/11·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.298552154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.298552154\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
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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
−9.273707276885932272852470470505, −9.141453528492942704595709107973, −8.595917921580745693787108849374, −7.85459564367845933724048728252, −7.68348048555598805857180133237, −7.46594192810939365406692402198, −6.81770418813623450899480633344, −6.70656156337577338450575889374, −6.31755624275359754380337576290, −5.83998847462150188180702483096, −5.49245375669920536201161866018, −5.15313189926702788425457926705, −4.26901777903278307552380391079, −3.80456597956703903027291671532, −3.42504147038113013085880278452, −3.19944623222774795090184095586, −2.22212161856329540190229865614, −1.98091267472640435563170042041, −1.54642921574642177238387606234, −0.816034237419382064912545105533,
0.816034237419382064912545105533, 1.54642921574642177238387606234, 1.98091267472640435563170042041, 2.22212161856329540190229865614, 3.19944623222774795090184095586, 3.42504147038113013085880278452, 3.80456597956703903027291671532, 4.26901777903278307552380391079, 5.15313189926702788425457926705, 5.49245375669920536201161866018, 5.83998847462150188180702483096, 6.31755624275359754380337576290, 6.70656156337577338450575889374, 6.81770418813623450899480633344, 7.46594192810939365406692402198, 7.68348048555598805857180133237, 7.85459564367845933724048728252, 8.595917921580745693787108849374, 9.141453528492942704595709107973, 9.273707276885932272852470470505
