L(s) = 1 | − 3·5-s − 4·9-s − 2·13-s + 7·17-s − 3·25-s − 6·29-s + 2·37-s + 9·41-s + 12·45-s + 6·49-s + 53-s − 6·61-s + 6·65-s + 27·73-s + 7·81-s − 21·85-s + 3·89-s − 18·97-s − 13·101-s − 21·109-s + 39·113-s + 8·117-s + 15·121-s + 30·125-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | − 1.34·5-s − 4/3·9-s − 0.554·13-s + 1.69·17-s − 3/5·25-s − 1.11·29-s + 0.328·37-s + 1.40·41-s + 1.78·45-s + 6/7·49-s + 0.137·53-s − 0.768·61-s + 0.744·65-s + 3.16·73-s + 7/9·81-s − 2.27·85-s + 0.317·89-s − 1.82·97-s − 1.29·101-s − 2.01·109-s + 3.66·113-s + 0.739·117-s + 1.36·121-s + 2.68·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 51 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 11 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 129 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.917939833095993676533297660225, −7.65010591666647941866464963891, −7.40053768627103976407539242648, −6.79035167899595959795658746889, −6.07445796535013780032241986773, −5.69560299467617460234089983612, −5.44206114614174896713964280730, −4.77985994314594267077533489697, −4.15778174628688710730491450011, −3.68623384396250968546785531482, −3.34454043013029617320640300353, −2.68328570602431600063470499629, −2.07462866304559559006807273816, −0.902507847438035719106670432727, 0,
0.902507847438035719106670432727, 2.07462866304559559006807273816, 2.68328570602431600063470499629, 3.34454043013029617320640300353, 3.68623384396250968546785531482, 4.15778174628688710730491450011, 4.77985994314594267077533489697, 5.44206114614174896713964280730, 5.69560299467617460234089983612, 6.07445796535013780032241986773, 6.79035167899595959795658746889, 7.40053768627103976407539242648, 7.65010591666647941866464963891, 7.917939833095993676533297660225