| L(s) = 1 | − 3-s + 4-s − 4·7-s + 9-s − 12-s − 2·13-s + 16-s − 4·19-s + 4·21-s + 25-s − 27-s − 4·28-s + 8·31-s + 36-s + 12·37-s + 2·39-s + 8·43-s − 48-s − 2·49-s − 2·52-s + 4·57-s − 4·61-s − 4·63-s + 64-s − 16·67-s − 75-s − 4·76-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/2·4-s − 1.51·7-s + 1/3·9-s − 0.288·12-s − 0.554·13-s + 1/4·16-s − 0.917·19-s + 0.872·21-s + 1/5·25-s − 0.192·27-s − 0.755·28-s + 1.43·31-s + 1/6·36-s + 1.97·37-s + 0.320·39-s + 1.21·43-s − 0.144·48-s − 2/7·49-s − 0.277·52-s + 0.529·57-s − 0.512·61-s − 0.503·63-s + 1/8·64-s − 1.95·67-s − 0.115·75-s − 0.458·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−8.402306645073389208875709046616, −7.61851966333210959386324957870, −7.50072014239365599576466271951, −6.77872728355853698929699351273, −6.49330656599515595141207552224, −5.96022304844885243033851457877, −5.92218145052084868507888987744, −4.99762985109706213804877258036, −4.39848825034532417015468798767, −4.13007375875743923718905749312, −3.12318249922503105552823930899, −2.87565984705891767205138792492, −2.18442355690526819133910690087, −1.10731363493759878565508314296, 0,
1.10731363493759878565508314296, 2.18442355690526819133910690087, 2.87565984705891767205138792492, 3.12318249922503105552823930899, 4.13007375875743923718905749312, 4.39848825034532417015468798767, 4.99762985109706213804877258036, 5.92218145052084868507888987744, 5.96022304844885243033851457877, 6.49330656599515595141207552224, 6.77872728355853698929699351273, 7.50072014239365599576466271951, 7.61851966333210959386324957870, 8.402306645073389208875709046616
