| L(s) = 1 | − 3-s − 8·7-s + 9-s − 4·13-s − 8·19-s + 8·21-s − 4·25-s − 27-s − 12·31-s − 6·37-s + 4·39-s + 4·43-s + 34·49-s + 8·57-s − 8·61-s − 8·63-s − 4·67-s − 14·73-s + 4·75-s − 16·79-s + 81-s + 32·91-s + 12·93-s + 10·97-s + 20·103-s − 6·109-s + 6·111-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 3.02·7-s + 1/3·9-s − 1.10·13-s − 1.83·19-s + 1.74·21-s − 4/5·25-s − 0.192·27-s − 2.15·31-s − 0.986·37-s + 0.640·39-s + 0.609·43-s + 34/7·49-s + 1.05·57-s − 1.02·61-s − 1.00·63-s − 0.488·67-s − 1.63·73-s + 0.461·75-s − 1.80·79-s + 1/9·81-s + 3.35·91-s + 1.24·93-s + 1.01·97-s + 1.97·103-s − 0.574·109-s + 0.569·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 292032 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 292032 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 76 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 96 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 88 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.639631547642643641883773363479, −7.65123434128088069321593738106, −7.24216625036064930106912054935, −6.98973896289571926370745539080, −6.33532890112439603644454312154, −6.10501472385236404646028041937, −5.72422818570997538151703885863, −5.01195645882144618631596131036, −4.24802253257068384608403362388, −3.80030406163941982702472675407, −3.24738621519568989285539207692, −2.64046204944647381129325887664, −1.89409231623715613149631655399, 0, 0,
1.89409231623715613149631655399, 2.64046204944647381129325887664, 3.24738621519568989285539207692, 3.80030406163941982702472675407, 4.24802253257068384608403362388, 5.01195645882144618631596131036, 5.72422818570997538151703885863, 6.10501472385236404646028041937, 6.33532890112439603644454312154, 6.98973896289571926370745539080, 7.24216625036064930106912054935, 7.65123434128088069321593738106, 8.639631547642643641883773363479
