| L(s) = 1 | − 2-s + 4-s − 3·5-s − 8-s + 3·10-s + 16-s + 4·19-s − 3·20-s − 3·23-s − 25-s + 3·29-s − 32-s − 4·38-s + 3·40-s − 2·43-s + 3·46-s − 15·47-s − 4·49-s + 50-s + 9·53-s − 3·58-s + 64-s + 67-s + 9·71-s − 14·73-s + 4·76-s − 3·80-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.353·8-s + 0.948·10-s + 1/4·16-s + 0.917·19-s − 0.670·20-s − 0.625·23-s − 1/5·25-s + 0.557·29-s − 0.176·32-s − 0.648·38-s + 0.474·40-s − 0.304·43-s + 0.442·46-s − 2.18·47-s − 4/7·49-s + 0.141·50-s + 1.23·53-s − 0.393·58-s + 1/8·64-s + 0.122·67-s + 1.06·71-s − 1.63·73-s + 0.458·76-s − 0.335·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 104 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.609029245222558289386495025739, −8.671234007572873596015962542049, −8.341906752583945825236596197607, −7.976681843896829700657710194487, −7.47078170298462966312389980291, −7.06531632154498006005389978897, −6.45668678671408005985928016769, −5.85877624435329572111912165516, −5.15610713357840073950821402514, −4.51935115568100078770355546120, −3.78610675719701428031694726911, −3.35211510701703793642784579063, −2.48351851344506650792229923687, −1.38051238992515136747189719480, 0,
1.38051238992515136747189719480, 2.48351851344506650792229923687, 3.35211510701703793642784579063, 3.78610675719701428031694726911, 4.51935115568100078770355546120, 5.15610713357840073950821402514, 5.85877624435329572111912165516, 6.45668678671408005985928016769, 7.06531632154498006005389978897, 7.47078170298462966312389980291, 7.976681843896829700657710194487, 8.341906752583945825236596197607, 8.671234007572873596015962542049, 9.609029245222558289386495025739
