| L(s) = 1 | − 2·5-s − 9-s + 2·13-s − 2·17-s + 3·25-s + 2·29-s − 2·41-s + 2·45-s + 4·49-s − 2·53-s − 2·61-s − 4·65-s + 2·73-s + 81-s + 4·85-s − 2·89-s − 2·97-s + 2·101-s + 2·109-s − 2·117-s + 4·121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + ⋯ |
| L(s) = 1 | − 2·5-s − 9-s + 2·13-s − 2·17-s + 3·25-s + 2·29-s − 2·41-s + 2·45-s + 4·49-s − 2·53-s − 2·61-s − 4·65-s + 2·73-s + 81-s + 4·85-s − 2·89-s − 2·97-s + 2·101-s + 2·109-s − 2·117-s + 4·121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5323935134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5323935134\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
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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
−7.29342028013455314973685785391, −6.75182038799543721294901214089, −6.72814245635822106468757900617, −6.72552270229868958113104509537, −6.18390062174837916117022491537, −6.18204788887165868167193491315, −6.00095206692150054700176195975, −5.66622558563739064601229591529, −5.30924291287216351748220960892, −5.10737313499839080746870562739, −4.71071284471956799887229606276, −4.68661615658372845467334466105, −4.51414379946950718010219316691, −4.05246587633755526302103191773, −3.98271308439624285514057492539, −3.74725239184903231767558580199, −3.39141442100870189581158695675, −3.21647588991003675215054211196, −3.01058750954397814600227403540, −2.57476138032715204162864717431, −2.46690302912986844679491042576, −1.97461444169619673993998852549, −1.50863776185989750114429940837, −1.10050280817860589540909830881, −0.59137151464952032825805018407,
0.59137151464952032825805018407, 1.10050280817860589540909830881, 1.50863776185989750114429940837, 1.97461444169619673993998852549, 2.46690302912986844679491042576, 2.57476138032715204162864717431, 3.01058750954397814600227403540, 3.21647588991003675215054211196, 3.39141442100870189581158695675, 3.74725239184903231767558580199, 3.98271308439624285514057492539, 4.05246587633755526302103191773, 4.51414379946950718010219316691, 4.68661615658372845467334466105, 4.71071284471956799887229606276, 5.10737313499839080746870562739, 5.30924291287216351748220960892, 5.66622558563739064601229591529, 6.00095206692150054700176195975, 6.18204788887165868167193491315, 6.18390062174837916117022491537, 6.72552270229868958113104509537, 6.72814245635822106468757900617, 6.75182038799543721294901214089, 7.29342028013455314973685785391
