| L(s) = 1 | + 12·5-s − 2·9-s + 10·13-s − 6·17-s + 70·25-s + 6·29-s + 22·37-s − 18·41-s − 24·45-s − 10·49-s + 6·61-s + 120·65-s + 9·81-s − 72·85-s − 24·89-s + 48·97-s + 66·101-s − 40·109-s − 18·113-s − 20·117-s + 10·121-s + 240·125-s + 127-s + 131-s + 137-s + 139-s + 72·145-s + ⋯ |
| L(s) = 1 | + 5.36·5-s − 2/3·9-s + 2.77·13-s − 1.45·17-s + 14·25-s + 1.11·29-s + 3.61·37-s − 2.81·41-s − 3.57·45-s − 1.42·49-s + 0.768·61-s + 14.8·65-s + 81-s − 7.80·85-s − 2.54·89-s + 4.87·97-s + 6.56·101-s − 3.83·109-s − 1.69·113-s − 1.84·117-s + 0.909·121-s + 21.4·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.97·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.28034148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.28034148\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| good | 3 | $C_2^3$ | \( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 7 | $C_2^3$ | \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 23 | $C_2^3$ | \( 1 + 2 T^{2} - 525 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 61 T^{2} + p^{2} T^{4} )( 1 + 83 T^{2} + p^{2} T^{4} ) \) |
| 47 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 41 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 106 T^{2} + 7755 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 3 T + 64 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 86 T^{2} + 2907 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^3$ | \( 1 + 106 T^{2} + 6195 T^{4} + 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )^{2}( 1 - 5 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.04187050123495732505650361291, −6.89096421369472557559618420237, −6.50615787271399208738507079661, −6.48470779602025264408663185356, −6.36624380210407394217537853300, −6.03141076952646877595223168477, −5.94509160426653465885009543959, −5.91202368558215649598219805058, −5.83608773451308238364373758240, −5.15406000223628009449234090817, −5.10686273703375726937881321658, −4.93535309589584820946809917498, −4.78957187865614652179639619191, −4.22689262766705898670740590399, −3.93592406260603035914177269084, −3.52160472589566567952065896370, −3.33180932208966774494329845747, −2.83992475314975998045867389890, −2.55975754618398702944466960711, −2.36709004255334101646557394849, −2.17648547397391964082529955453, −1.76872828528182976199289387104, −1.53632524360471275931762260685, −1.29004654483349892036376943181, −0.854807221390718308225457398643,
0.854807221390718308225457398643, 1.29004654483349892036376943181, 1.53632524360471275931762260685, 1.76872828528182976199289387104, 2.17648547397391964082529955453, 2.36709004255334101646557394849, 2.55975754618398702944466960711, 2.83992475314975998045867389890, 3.33180932208966774494329845747, 3.52160472589566567952065896370, 3.93592406260603035914177269084, 4.22689262766705898670740590399, 4.78957187865614652179639619191, 4.93535309589584820946809917498, 5.10686273703375726937881321658, 5.15406000223628009449234090817, 5.83608773451308238364373758240, 5.91202368558215649598219805058, 5.94509160426653465885009543959, 6.03141076952646877595223168477, 6.36624380210407394217537853300, 6.48470779602025264408663185356, 6.50615787271399208738507079661, 6.89096421369472557559618420237, 7.04187050123495732505650361291
