| L(s) = 1 | + 2·5-s − 2·9-s + 8·13-s − 6·17-s − 7·25-s + 4·29-s − 20·37-s + 20·41-s − 4·45-s − 5·49-s + 8·53-s + 26·61-s + 16·65-s + 18·73-s − 5·81-s − 12·85-s + 24·89-s − 16·97-s + 20·101-s − 20·113-s − 16·117-s + 3·121-s − 26·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2/3·9-s + 2.21·13-s − 1.45·17-s − 7/5·25-s + 0.742·29-s − 3.28·37-s + 3.12·41-s − 0.596·45-s − 5/7·49-s + 1.09·53-s + 3.32·61-s + 1.98·65-s + 2.10·73-s − 5/9·81-s − 1.30·85-s + 2.54·89-s − 1.62·97-s + 1.99·101-s − 1.88·113-s − 1.47·117-s + 3/11·121-s − 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.415391886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.415391886\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−8.121865683855346250917871835791, −7.46802549489449929304458830547, −6.89999375237673531400272497166, −6.39486340421971046323119110374, −6.35171638937504994192666881666, −5.61135260065455155907566257543, −5.55162300458149510222680140561, −4.93085486739421392203871722376, −4.17947794613570531538770424508, −3.73571330395388729025783799541, −3.51647821375355615052320307776, −2.48496537170117614857619865014, −2.22354879090908877160738042469, −1.55214010532178794455093749689, −0.67525660115678139253233977173,
0.67525660115678139253233977173, 1.55214010532178794455093749689, 2.22354879090908877160738042469, 2.48496537170117614857619865014, 3.51647821375355615052320307776, 3.73571330395388729025783799541, 4.17947794613570531538770424508, 4.93085486739421392203871722376, 5.55162300458149510222680140561, 5.61135260065455155907566257543, 6.35171638937504994192666881666, 6.39486340421971046323119110374, 6.89999375237673531400272497166, 7.46802549489449929304458830547, 8.121865683855346250917871835791
