| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 2·5-s + 4·6-s − 2·7-s − 4·8-s + 3·9-s − 4·10-s − 8·11-s − 6·12-s + 2·13-s + 4·14-s − 4·15-s + 5·16-s − 6·18-s − 2·19-s + 6·20-s + 4·21-s + 16·22-s − 2·23-s + 8·24-s + 3·25-s − 4·26-s − 4·27-s − 6·28-s − 2·29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s − 1.26·10-s − 2.41·11-s − 1.73·12-s + 0.554·13-s + 1.06·14-s − 1.03·15-s + 5/4·16-s − 1.41·18-s − 0.458·19-s + 1.34·20-s + 0.872·21-s + 3.41·22-s − 0.417·23-s + 1.63·24-s + 3/5·25-s − 0.784·26-s − 0.769·27-s − 1.13·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23328900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23328900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5231653343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5231653343\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 26 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 110 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 14 T + 122 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 22 T + 254 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 226 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 210 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.362967599396316486829863472358, −8.258056507400727254376860451563, −7.63434123589094092818093538131, −7.47372239932868302631239439214, −6.98776927958708483769091768969, −6.74454345873079194795529448422, −6.17685834616581263875829805446, −6.10507172763369240806262998574, −5.55558769398112576227433633561, −5.48948211358605629232458166069, −4.77626346732322801057939119861, −4.76968200226925766138049473349, −3.70550900179176380199755998901, −3.54531849207073823128693323875, −2.82252417428467463945743014812, −2.46715979681007685425053591565, −2.01027287291550629489391636555, −1.63168338707558880328145729319, −0.71065407895484290259084053375, −0.39273185389859996606138399586,
0.39273185389859996606138399586, 0.71065407895484290259084053375, 1.63168338707558880328145729319, 2.01027287291550629489391636555, 2.46715979681007685425053591565, 2.82252417428467463945743014812, 3.54531849207073823128693323875, 3.70550900179176380199755998901, 4.76968200226925766138049473349, 4.77626346732322801057939119861, 5.48948211358605629232458166069, 5.55558769398112576227433633561, 6.10507172763369240806262998574, 6.17685834616581263875829805446, 6.74454345873079194795529448422, 6.98776927958708483769091768969, 7.47372239932868302631239439214, 7.63434123589094092818093538131, 8.258056507400727254376860451563, 8.362967599396316486829863472358
