| L(s) = 1 | − 2-s − 3·3-s − 2·4-s + 3·6-s − 6·7-s + 3·8-s + 2·9-s − 6·11-s + 6·12-s − 3·13-s + 6·14-s + 16-s + 4·17-s − 2·18-s − 5·19-s + 18·21-s + 6·22-s + 2·23-s − 9·24-s + 3·26-s + 6·27-s + 12·28-s − 31-s − 2·32-s + 18·33-s − 4·34-s − 4·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 4-s + 1.22·6-s − 2.26·7-s + 1.06·8-s + 2/3·9-s − 1.80·11-s + 1.73·12-s − 0.832·13-s + 1.60·14-s + 1/4·16-s + 0.970·17-s − 0.471·18-s − 1.14·19-s + 3.92·21-s + 1.27·22-s + 0.417·23-s − 1.83·24-s + 0.588·26-s + 1.15·27-s + 2.26·28-s − 0.179·31-s − 0.353·32-s + 3.13·33-s − 0.685·34-s − 2/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 33 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 43 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + T + 33 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 7 T + 117 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 15 T + 163 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 + T + 91 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 21 T + 233 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 3 T + 137 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 163 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 162 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 203 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−12.93972281835560793328304327272, −12.77530826107366461194664135181, −12.05382558703418650096099764436, −11.73792435979112215104379094148, −10.86172981070757279343267880448, −10.26038123706751969123599255840, −9.986551135646300424046204636967, −9.939295121411511505824047360314, −8.780613974808470968139799786105, −8.647305319067773250473627408691, −7.72847099897749536043262283778, −6.98778454542719658500051129123, −6.42548266234641240966635633122, −5.83997881754905965570583860943, −5.10791352937727896511830478381, −4.94771465460436105819958449903, −3.59379870547119281719423165809, −2.81326367837647123037569197004, 0, 0,
2.81326367837647123037569197004, 3.59379870547119281719423165809, 4.94771465460436105819958449903, 5.10791352937727896511830478381, 5.83997881754905965570583860943, 6.42548266234641240966635633122, 6.98778454542719658500051129123, 7.72847099897749536043262283778, 8.647305319067773250473627408691, 8.780613974808470968139799786105, 9.939295121411511505824047360314, 9.986551135646300424046204636967, 10.26038123706751969123599255840, 10.86172981070757279343267880448, 11.73792435979112215104379094148, 12.05382558703418650096099764436, 12.77530826107366461194664135181, 12.93972281835560793328304327272
