| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 4·6-s − 2·7-s − 4·8-s + 2·9-s + 2·11-s + 6·12-s + 2·13-s + 4·14-s + 5·16-s + 4·17-s − 4·18-s − 10·19-s − 4·21-s − 4·22-s − 8·23-s − 8·24-s − 4·26-s + 6·27-s − 6·28-s + 4·31-s − 6·32-s + 4·33-s − 8·34-s + 6·36-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s − 0.755·7-s − 1.41·8-s + 2/3·9-s + 0.603·11-s + 1.73·12-s + 0.554·13-s + 1.06·14-s + 5/4·16-s + 0.970·17-s − 0.942·18-s − 2.29·19-s − 0.872·21-s − 0.852·22-s − 1.66·23-s − 1.63·24-s − 0.784·26-s + 1.15·27-s − 1.13·28-s + 0.718·31-s − 1.06·32-s + 0.696·33-s − 1.37·34-s + 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.799044806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.799044806\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 238 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.684648638717238585121319377801, −8.407142774775015435715800159721, −7.900058836222925737259314526788, −7.896314873822675993737841246852, −7.43282382536693662645398700543, −6.83134186277352371810084411107, −6.58745766493358263861563444766, −6.26833104785699265051346445140, −5.89549374886094321173671841092, −5.65998015200224339991499999846, −4.72855063337205151094649498891, −4.27837102898916586143532318742, −3.87976465877525901154334684993, −3.66176082140666523818778325737, −2.81501425497379130970789188319, −2.78865560584124379612603378894, −2.14058999202614419691078028552, −1.80418779913884958590731653189, −1.05460069688527135846587227707, −0.51124012064741364046290632354,
0.51124012064741364046290632354, 1.05460069688527135846587227707, 1.80418779913884958590731653189, 2.14058999202614419691078028552, 2.78865560584124379612603378894, 2.81501425497379130970789188319, 3.66176082140666523818778325737, 3.87976465877525901154334684993, 4.27837102898916586143532318742, 4.72855063337205151094649498891, 5.65998015200224339991499999846, 5.89549374886094321173671841092, 6.26833104785699265051346445140, 6.58745766493358263861563444766, 6.83134186277352371810084411107, 7.43282382536693662645398700543, 7.896314873822675993737841246852, 7.900058836222925737259314526788, 8.407142774775015435715800159721, 8.684648638717238585121319377801
