L(s) = 1 | + 3·4-s + 2·5-s − 3·9-s + 5·16-s + 6·20-s + 10·23-s + 3·25-s + 14·31-s − 9·36-s + 10·37-s − 6·45-s − 2·47-s + 4·49-s + 6·53-s + 6·59-s + 3·64-s + 4·67-s + 4·71-s + 10·80-s + 9·81-s − 18·89-s + 30·92-s + 9·100-s + 32·103-s − 26·113-s + 20·115-s + 42·124-s + ⋯ |
L(s) = 1 | + 3/2·4-s + 0.894·5-s − 9-s + 5/4·16-s + 1.34·20-s + 2.08·23-s + 3/5·25-s + 2.51·31-s − 3/2·36-s + 1.64·37-s − 0.894·45-s − 0.291·47-s + 4/7·49-s + 0.824·53-s + 0.781·59-s + 3/8·64-s + 0.488·67-s + 0.474·71-s + 1.11·80-s + 81-s − 1.90·89-s + 3.12·92-s + 9/10·100-s + 3.15·103-s − 2.44·113-s + 1.86·115-s + 3.77·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.977909085\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.977909085\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.27235551798184107567472542307, −7.12751031327383728769576247541, −6.60677053114774298869344297105, −6.21701070809685125162297414872, −6.05862890852302505081937052618, −5.56295493704910345932800593174, −4.97612336232248743235362735232, −4.79579168740900751026542052888, −3.99761165840674192147871025147, −3.33609464081708860857704629528, −2.80674013060930219623577924836, −2.54314044356119275812107443124, −2.28175397030153669658824547077, −1.27205112463003282085915637256, −0.906207109284252358515751529287,
0.906207109284252358515751529287, 1.27205112463003282085915637256, 2.28175397030153669658824547077, 2.54314044356119275812107443124, 2.80674013060930219623577924836, 3.33609464081708860857704629528, 3.99761165840674192147871025147, 4.79579168740900751026542052888, 4.97612336232248743235362735232, 5.56295493704910345932800593174, 6.05862890852302505081937052618, 6.21701070809685125162297414872, 6.60677053114774298869344297105, 7.12751031327383728769576247541, 7.27235551798184107567472542307