| L(s) = 1 | − 2-s − 3-s − 2·4-s + 5-s + 6-s − 2·7-s + 3·8-s − 4·9-s − 10-s − 2·11-s + 2·12-s − 8·13-s + 2·14-s − 15-s + 16-s − 4·17-s + 4·18-s − 19-s − 2·20-s + 2·21-s + 2·22-s − 5·23-s − 3·24-s − 8·25-s + 8·26-s + 6·27-s + 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s + 1.06·8-s − 4/3·9-s − 0.316·10-s − 0.603·11-s + 0.577·12-s − 2.21·13-s + 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.942·18-s − 0.229·19-s − 0.447·20-s + 0.436·21-s + 0.426·22-s − 1.04·23-s − 0.612·24-s − 8/5·25-s + 1.56·26-s + 1.15·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82369 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82369 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 41 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 27 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 51 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 53 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 5 T + 37 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 7 T + 117 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 19 T + 207 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 127 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 13 T + 145 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + p T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 67 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 147 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 3 T + 119 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 15 T + 149 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
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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
−11.63546616106398495353504171243, −11.08267367468802642398819168410, −10.22720194621860615949496416826, −10.19800128145452940706623265426, −9.705977787948646897032949631316, −9.359695630173813234106596060774, −8.639427512439216648925870758473, −8.489232094052008884558852424876, −7.85168181858551961766210150419, −7.19490164992474193196864611389, −6.73666697600104912432448076844, −5.96012666303608533283652606899, −5.34549592369674859869061891994, −5.29805219343833053366075221404, −4.35823504622890002555233087633, −3.81217979207927878359753821501, −2.62714397855180607672535516985, −2.23799494825827285624372484241, 0, 0,
2.23799494825827285624372484241, 2.62714397855180607672535516985, 3.81217979207927878359753821501, 4.35823504622890002555233087633, 5.29805219343833053366075221404, 5.34549592369674859869061891994, 5.96012666303608533283652606899, 6.73666697600104912432448076844, 7.19490164992474193196864611389, 7.85168181858551961766210150419, 8.489232094052008884558852424876, 8.639427512439216648925870758473, 9.359695630173813234106596060774, 9.705977787948646897032949631316, 10.19800128145452940706623265426, 10.22720194621860615949496416826, 11.08267367468802642398819168410, 11.63546616106398495353504171243
