| L(s) = 1 | − 2-s + 3-s + 4-s − 4·5-s − 6-s − 8-s + 9-s + 4·10-s + 12-s − 4·15-s + 16-s − 18-s − 8·19-s − 4·20-s + 2·23-s − 24-s + 11·25-s + 27-s + 10·29-s + 4·30-s − 32-s + 36-s + 8·38-s + 4·40-s − 2·43-s − 4·45-s − 2·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.288·12-s − 1.03·15-s + 1/4·16-s − 0.235·18-s − 1.83·19-s − 0.894·20-s + 0.417·23-s − 0.204·24-s + 11/5·25-s + 0.192·27-s + 1.85·29-s + 0.730·30-s − 0.176·32-s + 1/6·36-s + 1.29·38-s + 0.632·40-s − 0.304·43-s − 0.596·45-s − 0.294·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−9.339567424682786430784453915247, −8.673694079790885226677238103851, −8.362799533606162582050264738401, −8.159133907905646093204816135301, −7.60740765702098772208639847156, −6.99699840001464631994095197982, −6.59540810342012874523421779116, −6.11652543717961444714848438316, −4.81772000759950662932311022970, −4.63656510970371918111347666082, −3.88273063652071506820021780492, −3.20886576284072801053693960150, −2.66417888904161937297868600496, −1.46767016993003419922605384526, 0,
1.46767016993003419922605384526, 2.66417888904161937297868600496, 3.20886576284072801053693960150, 3.88273063652071506820021780492, 4.63656510970371918111347666082, 4.81772000759950662932311022970, 6.11652543717961444714848438316, 6.59540810342012874523421779116, 6.99699840001464631994095197982, 7.60740765702098772208639847156, 8.159133907905646093204816135301, 8.362799533606162582050264738401, 8.673694079790885226677238103851, 9.339567424682786430784453915247
