| L(s) = 1 | − 2·3-s + 9-s + 6·11-s + 3·13-s + 2·25-s + 4·27-s − 12·33-s − 8·37-s − 6·39-s + 6·47-s + 2·49-s − 18·59-s − 20·61-s − 6·71-s + 16·73-s − 4·75-s − 11·81-s − 6·83-s + 4·97-s + 6·99-s − 12·107-s + 4·109-s + 16·111-s + 3·117-s + 14·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s + 1.80·11-s + 0.832·13-s + 2/5·25-s + 0.769·27-s − 2.08·33-s − 1.31·37-s − 0.960·39-s + 0.875·47-s + 2/7·49-s − 2.34·59-s − 2.56·61-s − 0.712·71-s + 1.87·73-s − 0.461·75-s − 1.22·81-s − 0.658·83-s + 0.406·97-s + 0.603·99-s − 1.16·107-s + 0.383·109-s + 1.51·111-s + 0.277·117-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7067454387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7067454387\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 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$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 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
−11.87155116813077068640664381906, −11.22223741775793504704253476270, −10.81731167510890640934599763781, −10.37515220498659616000734567466, −9.408432747001542418489178532692, −9.028704331668933074461162446989, −8.456296581918437833902757190517, −7.50956221473763168283760347313, −6.76577819406238587413747469202, −6.27901981478651695572674737545, −5.80690615762205488723594748732, −4.88918895983617812429598939925, −4.14856945798263111009867585329, −3.24989607732100239535744615059, −1.41770899276333058782442932324,
1.41770899276333058782442932324, 3.24989607732100239535744615059, 4.14856945798263111009867585329, 4.88918895983617812429598939925, 5.80690615762205488723594748732, 6.27901981478651695572674737545, 6.76577819406238587413747469202, 7.50956221473763168283760347313, 8.456296581918437833902757190517, 9.028704331668933074461162446989, 9.408432747001542418489178532692, 10.37515220498659616000734567466, 10.81731167510890640934599763781, 11.22223741775793504704253476270, 11.87155116813077068640664381906
