L(s) = 1 | + 4-s − 8·7-s + 9-s + 4·13-s + 16-s + 12·17-s + 25-s − 8·28-s − 12·29-s + 36-s + 4·37-s − 8·43-s + 34·49-s + 4·52-s − 6·53-s − 8·63-s + 64-s + 12·68-s + 81-s + 36·89-s − 32·91-s + 4·97-s + 100-s − 24·107-s − 8·112-s − 36·113-s − 12·116-s + ⋯ |
L(s) = 1 | + 1/2·4-s − 3.02·7-s + 1/3·9-s + 1.10·13-s + 1/4·16-s + 2.91·17-s + 1/5·25-s − 1.51·28-s − 2.22·29-s + 1/6·36-s + 0.657·37-s − 1.21·43-s + 34/7·49-s + 0.554·52-s − 0.824·53-s − 1.00·63-s + 1/8·64-s + 1.45·68-s + 1/9·81-s + 3.81·89-s − 3.35·91-s + 0.406·97-s + 1/10·100-s − 2.32·107-s − 0.755·112-s − 3.38·113-s − 1.11·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2528100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2528100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 53 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 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
−7.40299975179098898623930398115, −6.93165528247052272579764875319, −6.42217617652666865799421983967, −6.29699741617005267930031156094, −5.92460482285255092724220602337, −5.37985442259477011829539614630, −5.16033961728710776108999214836, −3.92189663406678546150391342843, −3.78458604271642354350170110042, −3.36585804145949552210873743484, −3.11065830387495411526913738527, −2.55325351537140642139443836766, −1.59750473666486892284412590277, −0.978287305185245428011662937257, 0,
0.978287305185245428011662937257, 1.59750473666486892284412590277, 2.55325351537140642139443836766, 3.11065830387495411526913738527, 3.36585804145949552210873743484, 3.78458604271642354350170110042, 3.92189663406678546150391342843, 5.16033961728710776108999214836, 5.37985442259477011829539614630, 5.92460482285255092724220602337, 6.29699741617005267930031156094, 6.42217617652666865799421983967, 6.93165528247052272579764875319, 7.40299975179098898623930398115