| L(s) = 1 | + 4·3-s + 10·9-s − 8·13-s + 4·16-s + 12·17-s + 6·23-s − 9·25-s + 32·27-s + 12·29-s − 32·39-s + 4·43-s + 16·48-s + 48·51-s + 18·53-s + 16·61-s + 24·69-s − 36·75-s + 18·79-s + 89·81-s + 48·87-s + 12·101-s + 24·107-s − 60·113-s − 80·117-s − 18·121-s + 127-s + 16·129-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 10/3·9-s − 2.21·13-s + 16-s + 2.91·17-s + 1.25·23-s − 9/5·25-s + 6.15·27-s + 2.22·29-s − 5.12·39-s + 0.609·43-s + 2.30·48-s + 6.72·51-s + 2.47·53-s + 2.04·61-s + 2.88·69-s − 4.15·75-s + 2.02·79-s + 89/9·81-s + 5.14·87-s + 1.19·101-s + 2.32·107-s − 5.64·113-s − 7.39·117-s − 1.63·121-s + 0.0887·127-s + 1.40·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.899355478\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.899355478\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) |
| 3 | $C_2^2$ | \( ( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^3$ | \( 1 + 9 T^{2} + 56 T^{4} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 18 T^{2} + 203 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 + 29 T^{2} + 480 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 + 53 T^{2} + 1848 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 + 38 T^{2} + 75 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 27 T^{2} - 1480 T^{4} - 27 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 54 T^{2} - 565 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 10 T^{2} - 4389 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 + 65 T^{2} - 1104 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 45 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 153 T^{2} + 15488 T^{4} + 153 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 113 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.69886359132345263770259541458, −7.66472001774795450148252438123, −7.24512420702448550969365528245, −6.99107251461381933806022610827, −6.87379725261714268445240803608, −6.61268433625115392644138889161, −6.23564539502329677663121703039, −6.05989271097388934422250962513, −5.43472064848008836252219137932, −5.42423301420494239919116561392, −5.17091526955420967882773165144, −4.84745256009241011484472340332, −4.75859115547961153425264867321, −4.36337050088606298719483544142, −3.84000591229597774409823339438, −3.65806054521207835139998591699, −3.64037635922941505596439581148, −3.25214137570193049920871467835, −2.74411017026177378895868327267, −2.63501726559528313866156655319, −2.40802780610440579685641170607, −2.35599495893713423354033459174, −1.42964512770912454141565688065, −1.04894841684322899871543530561, −0.976470815622892274161568489520,
0.976470815622892274161568489520, 1.04894841684322899871543530561, 1.42964512770912454141565688065, 2.35599495893713423354033459174, 2.40802780610440579685641170607, 2.63501726559528313866156655319, 2.74411017026177378895868327267, 3.25214137570193049920871467835, 3.64037635922941505596439581148, 3.65806054521207835139998591699, 3.84000591229597774409823339438, 4.36337050088606298719483544142, 4.75859115547961153425264867321, 4.84745256009241011484472340332, 5.17091526955420967882773165144, 5.42423301420494239919116561392, 5.43472064848008836252219137932, 6.05989271097388934422250962513, 6.23564539502329677663121703039, 6.61268433625115392644138889161, 6.87379725261714268445240803608, 6.99107251461381933806022610827, 7.24512420702448550969365528245, 7.66472001774795450148252438123, 7.69886359132345263770259541458
