| L(s) = 1 | + 2·4-s + 3·5-s − 7·7-s − 2·9-s + 6·20-s + 2·25-s − 3·27-s − 14·28-s − 29-s − 6·31-s − 21·35-s − 4·36-s + 5·43-s − 6·45-s + 25·49-s − 14·61-s + 14·63-s − 8·64-s + 7·71-s − 5·73-s + 4·81-s − 5·89-s − 17·97-s + 4·100-s − 6·108-s − 8·109-s + 12·113-s + ⋯ |
| L(s) = 1 | + 4-s + 1.34·5-s − 2.64·7-s − 2/3·9-s + 1.34·20-s + 2/5·25-s − 0.577·27-s − 2.64·28-s − 0.185·29-s − 1.07·31-s − 3.54·35-s − 2/3·36-s + 0.762·43-s − 0.894·45-s + 25/7·49-s − 1.79·61-s + 1.76·63-s − 64-s + 0.830·71-s − 0.585·73-s + 4/9·81-s − 0.529·89-s − 1.72·97-s + 2/5·100-s − 0.577·108-s − 0.766·109-s + 1.12·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178215 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178215 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 109 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 84 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.242606333032966863229388121402, −8.670080992065839130683426935279, −7.83687480289633459567071683610, −7.26581902099383651215235141337, −6.83820435081765263667733416479, −6.39843801951569783144542814058, −6.02040960730402598312747827511, −5.79249015734849650138681729728, −5.18745873006664154053979005610, −4.08146645959628896119151801503, −3.46946480213921829250591155224, −2.85012982960755850902706871727, −2.51005616545012661441284309563, −1.69057423116807441233189096306, 0,
1.69057423116807441233189096306, 2.51005616545012661441284309563, 2.85012982960755850902706871727, 3.46946480213921829250591155224, 4.08146645959628896119151801503, 5.18745873006664154053979005610, 5.79249015734849650138681729728, 6.02040960730402598312747827511, 6.39843801951569783144542814058, 6.83820435081765263667733416479, 7.26581902099383651215235141337, 7.83687480289633459567071683610, 8.670080992065839130683426935279, 9.242606333032966863229388121402
