| L(s) = 1 | − 2-s + 2·3-s − 2·4-s − 2·6-s + 7-s + 3·8-s + 3·9-s − 4·12-s + 3·13-s − 14-s + 16-s − 4·17-s − 3·18-s + 4·19-s + 2·21-s − 23-s + 6·24-s − 3·26-s + 4·27-s − 2·28-s − 4·29-s − 9·31-s − 2·32-s + 4·34-s − 6·36-s − 4·38-s + 6·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 4-s − 0.816·6-s + 0.377·7-s + 1.06·8-s + 9-s − 1.15·12-s + 0.832·13-s − 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.707·18-s + 0.917·19-s + 0.436·21-s − 0.208·23-s + 1.22·24-s − 0.588·26-s + 0.769·27-s − 0.377·28-s − 0.742·29-s − 1.61·31-s − 0.353·32-s + 0.685·34-s − 36-s − 0.648·38-s + 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 27 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 35 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 57 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9 T + 71 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + 51 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 74 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 11 T + 105 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 77 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 153 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 165 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 122 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 5 T + 161 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 13 T + 159 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−7.57475287483566565364646436900, −7.54532243659981522881496937388, −7.03656318914249980637845745231, −6.62069821216053124729633136333, −6.28669857378201183328779867833, −5.89651334799432474846262677999, −5.25572557331910140267308353940, −5.20851960109425546891864849878, −4.65673364785470440437951157562, −4.35774527107344163844783463491, −4.03798749105075112713196791113, −3.61370190195417911733283452317, −3.21583733705274006258453032334, −3.02103403806531565060216947168, −2.26156047156882474840053708483, −1.89691566294656990845772889442, −1.36369885284836127983785139961, −1.20174942982794437341791577984, 0, 0,
1.20174942982794437341791577984, 1.36369885284836127983785139961, 1.89691566294656990845772889442, 2.26156047156882474840053708483, 3.02103403806531565060216947168, 3.21583733705274006258453032334, 3.61370190195417911733283452317, 4.03798749105075112713196791113, 4.35774527107344163844783463491, 4.65673364785470440437951157562, 5.20851960109425546891864849878, 5.25572557331910140267308353940, 5.89651334799432474846262677999, 6.28669857378201183328779867833, 6.62069821216053124729633136333, 7.03656318914249980637845745231, 7.54532243659981522881496937388, 7.57475287483566565364646436900
