| L(s) = 1 | − 4·7-s − 2·19-s + 8·25-s − 12·29-s − 12·41-s + 8·43-s − 2·49-s − 12·53-s − 8·61-s − 24·71-s − 20·73-s − 12·89-s − 24·107-s + 12·113-s + 20·121-s + 127-s + 131-s + 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.458·19-s + 8/5·25-s − 2.22·29-s − 1.87·41-s + 1.21·43-s − 2/7·49-s − 1.64·53-s − 1.02·61-s − 2.84·71-s − 2.34·73-s − 1.27·89-s − 2.32·107-s + 1.12·113-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.693·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
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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
−8.601433684672441509933749686453, −8.503921129058680073655437423422, −7.67127216582801290875691661686, −7.55710854004184165992271798827, −6.94671831811922356897585384104, −6.90721777719929407468632365187, −6.15742851137146798289555343564, −6.14533025279164946267054485543, −5.65065048162430314843615952902, −5.16211025902678104156867854742, −4.56984495967064686692884362074, −4.40718883959655407034679644922, −3.52475650272781103264045855467, −3.51518442895303260884905164114, −2.86170632736332335582529912842, −2.62789003179984182275392760250, −1.66307943256145841616912654843, −1.38163295175431056396445053206, 0, 0,
1.38163295175431056396445053206, 1.66307943256145841616912654843, 2.62789003179984182275392760250, 2.86170632736332335582529912842, 3.51518442895303260884905164114, 3.52475650272781103264045855467, 4.40718883959655407034679644922, 4.56984495967064686692884362074, 5.16211025902678104156867854742, 5.65065048162430314843615952902, 6.14533025279164946267054485543, 6.15742851137146798289555343564, 6.90721777719929407468632365187, 6.94671831811922356897585384104, 7.55710854004184165992271798827, 7.67127216582801290875691661686, 8.503921129058680073655437423422, 8.601433684672441509933749686453
