| L(s) = 1 | − 4·4-s + 4·7-s − 8·13-s + 12·16-s − 2·19-s + 25-s − 16·28-s − 2·31-s + 16·37-s − 8·43-s − 2·49-s + 32·52-s − 20·61-s − 32·64-s + 28·67-s + 4·73-s + 8·76-s − 32·79-s − 32·91-s − 8·97-s − 4·100-s + 28·103-s − 2·109-s + 48·112-s − 13·121-s + 8·124-s + 127-s + ⋯ |
| L(s) = 1 | − 2·4-s + 1.51·7-s − 2.21·13-s + 3·16-s − 0.458·19-s + 1/5·25-s − 3.02·28-s − 0.359·31-s + 2.63·37-s − 1.21·43-s − 2/7·49-s + 4.43·52-s − 2.56·61-s − 4·64-s + 3.42·67-s + 0.468·73-s + 0.917·76-s − 3.60·79-s − 3.35·91-s − 0.812·97-s − 2/5·100-s + 2.75·103-s − 0.191·109-s + 4.53·112-s − 1.18·121-s + 0.718·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
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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
−8.922941609669290152495843868069, −8.538687880420415041094117740367, −8.027105501047314182413259178583, −7.73102144096255023104365276788, −7.36532580760380206733382868882, −6.47289908140934830552102870516, −5.75188631797463493014423176019, −5.18690236062207152876422818521, −4.76552433542106688375778775229, −4.59634975877424853850281958730, −4.04017076909632160868815003758, −3.14233849758258863442381605357, −2.30189805881882976209364768483, −1.27645214139714047950709921081, 0,
1.27645214139714047950709921081, 2.30189805881882976209364768483, 3.14233849758258863442381605357, 4.04017076909632160868815003758, 4.59634975877424853850281958730, 4.76552433542106688375778775229, 5.18690236062207152876422818521, 5.75188631797463493014423176019, 6.47289908140934830552102870516, 7.36532580760380206733382868882, 7.73102144096255023104365276788, 8.027105501047314182413259178583, 8.538687880420415041094117740367, 8.922941609669290152495843868069
