| L(s) = 1 | + 2·5-s + 8·19-s + 8·23-s − 25-s + 10·29-s + 8·49-s − 8·53-s − 8·67-s − 8·71-s − 2·73-s + 16·95-s + 2·97-s − 2·101-s + 16·115-s + 16·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 20·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 16·169-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.83·19-s + 1.66·23-s − 1/5·25-s + 1.85·29-s + 8/7·49-s − 1.09·53-s − 0.977·67-s − 0.949·71-s − 0.234·73-s + 1.64·95-s + 0.203·97-s − 0.199·101-s + 1.49·115-s + 1.45·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.66·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.23·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.641312771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.641312771\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
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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
−8.639171249230063486522798045557, −8.036143398792126485992129264280, −7.40341062264873504400436963094, −7.29215055815531988455327182918, −6.58892096714760052492517309456, −6.23129166402151517542951628468, −5.66833683606517424595655424300, −5.25759546296790927324314756629, −4.80964450312251885122672450735, −4.31322751094338848119674209615, −3.42139734069438610373225945433, −2.98626379292106237150725333962, −2.50558707717778465858325203762, −1.53339075285070896652215249768, −0.945009788358625776084727136125,
0.945009788358625776084727136125, 1.53339075285070896652215249768, 2.50558707717778465858325203762, 2.98626379292106237150725333962, 3.42139734069438610373225945433, 4.31322751094338848119674209615, 4.80964450312251885122672450735, 5.25759546296790927324314756629, 5.66833683606517424595655424300, 6.23129166402151517542951628468, 6.58892096714760052492517309456, 7.29215055815531988455327182918, 7.40341062264873504400436963094, 8.036143398792126485992129264280, 8.639171249230063486522798045557
