| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 2·7-s − 3·8-s + 3·9-s − 7·11-s − 2·12-s + 4·13-s + 2·14-s + 16-s + 6·17-s − 3·18-s + 2·19-s + 4·21-s + 7·22-s − 9·23-s + 6·24-s − 4·26-s − 4·27-s − 2·28-s + 2·29-s + 8·31-s + 32-s + 14·33-s − 6·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.755·7-s − 1.06·8-s + 9-s − 2.11·11-s − 0.577·12-s + 1.10·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.707·18-s + 0.458·19-s + 0.872·21-s + 1.49·22-s − 1.87·23-s + 1.22·24-s − 0.784·26-s − 0.769·27-s − 0.377·28-s + 0.371·29-s + 1.43·31-s + 0.176·32-s + 2.43·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8484326010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8484326010\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 9 T + 56 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 9 T + 64 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 18 T + 158 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 11 T + 132 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 174 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 128 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - T + 162 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 156 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−9.313222484854293335946781178968, −8.768828903644330276023609874055, −8.456027232849861413611735095652, −8.223774013429579429353856889744, −7.67028272507488677244890318114, −7.18347387849391872063882335077, −7.14973796147253365657969554588, −6.40092069596639209662367063643, −6.08519516462701222910325816287, −5.73218190932137707230511023857, −5.35367422485844545339458319311, −5.35043315509478588001394626005, −4.29007800986332889679917777341, −4.02152142549637783535081537172, −3.47490252229220580230117733943, −2.71097211105444074065876313499, −2.62268070207715116817454999560, −1.83965249723161771003265248928, −0.69959274062441458301279973544, −0.64227985210694428018150375061,
0.64227985210694428018150375061, 0.69959274062441458301279973544, 1.83965249723161771003265248928, 2.62268070207715116817454999560, 2.71097211105444074065876313499, 3.47490252229220580230117733943, 4.02152142549637783535081537172, 4.29007800986332889679917777341, 5.35043315509478588001394626005, 5.35367422485844545339458319311, 5.73218190932137707230511023857, 6.08519516462701222910325816287, 6.40092069596639209662367063643, 7.14973796147253365657969554588, 7.18347387849391872063882335077, 7.67028272507488677244890318114, 8.223774013429579429353856889744, 8.456027232849861413611735095652, 8.768828903644330276023609874055, 9.313222484854293335946781178968
