| L(s) = 1 | + 4·7-s + 3·13-s − 3·17-s + 5·19-s + 10·25-s + 3·29-s + 31-s − 7·37-s − 3·41-s + 3·43-s + 9·47-s + 9·49-s − 9·53-s + 15·59-s + 3·61-s + 27·67-s + 3·73-s − 3·79-s − 9·83-s − 3·89-s + 12·91-s + 3·97-s + 16·103-s − 3·107-s − 11·109-s − 9·113-s − 12·119-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 0.832·13-s − 0.727·17-s + 1.14·19-s + 2·25-s + 0.557·29-s + 0.179·31-s − 1.15·37-s − 0.468·41-s + 0.457·43-s + 1.31·47-s + 9/7·49-s − 1.23·53-s + 1.95·59-s + 0.384·61-s + 3.29·67-s + 0.351·73-s − 0.337·79-s − 0.987·83-s − 0.317·89-s + 1.25·91-s + 0.304·97-s + 1.57·103-s − 0.290·107-s − 1.05·109-s − 0.846·113-s − 1.10·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.518349424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.518349424\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T + 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 15 T + 166 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 3 T + 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 3 T + 82 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 3 T + 92 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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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
−8.759039626642297413233620604758, −8.536271032411513886718532306160, −8.150851192489077845958268523311, −8.036147384499900609775357932966, −7.27907668908426428047659470840, −7.06532452135961264595519351025, −6.75735802925065322513530812868, −6.38296985635137218168520784676, −5.63871213535822847130284441172, −5.47541256431122444025574166158, −5.00429859970751961609546886243, −4.80016310056391510687520256896, −4.12954129457499388208496369455, −3.98782490682425775861356552641, −3.08427961041977245396538751738, −3.07066876321427102921070974695, −2.08390278318428388577052818038, −1.95445396973836454044708895328, −0.935167044601348290200386842033, −0.921362689006216971261119078839,
0.921362689006216971261119078839, 0.935167044601348290200386842033, 1.95445396973836454044708895328, 2.08390278318428388577052818038, 3.07066876321427102921070974695, 3.08427961041977245396538751738, 3.98782490682425775861356552641, 4.12954129457499388208496369455, 4.80016310056391510687520256896, 5.00429859970751961609546886243, 5.47541256431122444025574166158, 5.63871213535822847130284441172, 6.38296985635137218168520784676, 6.75735802925065322513530812868, 7.06532452135961264595519351025, 7.27907668908426428047659470840, 8.036147384499900609775357932966, 8.150851192489077845958268523311, 8.536271032411513886718532306160, 8.759039626642297413233620604758
