L(s) = 1 | + 4·3-s + 6·9-s + 12·17-s − 4·19-s − 10·25-s − 4·27-s + 12·41-s − 16·43-s + 49-s + 48·51-s − 16·57-s + 12·59-s + 8·67-s + 4·73-s − 40·75-s − 37·81-s + 12·83-s − 12·89-s − 20·97-s − 24·107-s + 12·113-s − 22·121-s + 48·123-s + 127-s − 64·129-s + 131-s + 137-s + ⋯ |
L(s) = 1 | + 2.30·3-s + 2·9-s + 2.91·17-s − 0.917·19-s − 2·25-s − 0.769·27-s + 1.87·41-s − 2.43·43-s + 1/7·49-s + 6.72·51-s − 2.11·57-s + 1.56·59-s + 0.977·67-s + 0.468·73-s − 4.61·75-s − 4.11·81-s + 1.31·83-s − 1.27·89-s − 2.03·97-s − 2.32·107-s + 1.12·113-s − 2·121-s + 4.32·123-s + 0.0887·127-s − 5.63·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.785560266\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.785560266\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−10.04491938039307927961472194001, −9.392494157732895222790142944171, −9.195409247501138715663586603039, −8.222985047096640302551257734717, −8.182643116839158569055135937831, −7.87021772021814993834479658639, −7.29668061464756137953129595214, −6.47781377385923123584313505983, −5.62168100149868099368912779374, −5.36663802373794330843690451326, −3.93522119575008455702519349301, −3.82418745638366191622617161116, −3.07302685938302839530976846531, −2.48514181668809031419660323366, −1.64950502134773404742708641242,
1.64950502134773404742708641242, 2.48514181668809031419660323366, 3.07302685938302839530976846531, 3.82418745638366191622617161116, 3.93522119575008455702519349301, 5.36663802373794330843690451326, 5.62168100149868099368912779374, 6.47781377385923123584313505983, 7.29668061464756137953129595214, 7.87021772021814993834479658639, 8.182643116839158569055135937831, 8.222985047096640302551257734717, 9.195409247501138715663586603039, 9.392494157732895222790142944171, 10.04491938039307927961472194001