L(s) = 1 | + 2·3-s + 9-s − 4·11-s + 8·13-s + 2·25-s − 4·27-s − 8·33-s + 8·37-s + 16·39-s + 16·47-s + 2·49-s − 4·59-s + 8·61-s − 16·71-s − 4·73-s + 4·75-s − 11·81-s + 20·83-s − 4·97-s − 4·99-s − 4·107-s + 8·109-s + 16·111-s + 8·117-s + 6·121-s + 127-s + 131-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1/3·9-s − 1.20·11-s + 2.21·13-s + 2/5·25-s − 0.769·27-s − 1.39·33-s + 1.31·37-s + 2.56·39-s + 2.33·47-s + 2/7·49-s − 0.520·59-s + 1.02·61-s − 1.89·71-s − 0.468·73-s + 0.461·75-s − 1.22·81-s + 2.19·83-s − 0.406·97-s − 0.402·99-s − 0.386·107-s + 0.766·109-s + 1.51·111-s + 0.739·117-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.463303538\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.463303538\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.157649257941150629627668744017, −8.605998086911184244027567386456, −8.563518665434638648441097602158, −7.889351246962474617109757741872, −7.56261637712722061197144203238, −7.01736426340708825263278794244, −6.12923638153892308020168201344, −5.94413479604384105205810824292, −5.32590511408509228443148681242, −4.49248161564823780437167120124, −3.92542181323834276976637053392, −3.38277261268489150385048433794, −2.76583292885254762784498307777, −2.17444951706439887887311187476, −1.07507801443192078105758943254,
1.07507801443192078105758943254, 2.17444951706439887887311187476, 2.76583292885254762784498307777, 3.38277261268489150385048433794, 3.92542181323834276976637053392, 4.49248161564823780437167120124, 5.32590511408509228443148681242, 5.94413479604384105205810824292, 6.12923638153892308020168201344, 7.01736426340708825263278794244, 7.56261637712722061197144203238, 7.889351246962474617109757741872, 8.563518665434638648441097602158, 8.605998086911184244027567386456, 9.157649257941150629627668744017