| L(s) = 1 | − 2·2-s + 3·4-s + 2·5-s − 5·7-s − 4·8-s − 4·10-s − 2·11-s − 7·13-s + 10·14-s + 5·16-s + 8·17-s − 5·19-s + 6·20-s + 4·22-s + 4·23-s + 5·25-s + 14·26-s − 15·28-s + 8·29-s − 6·32-s − 16·34-s − 10·35-s + 14·37-s + 10·38-s − 8·40-s + 10·41-s − 7·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.894·5-s − 1.88·7-s − 1.41·8-s − 1.26·10-s − 0.603·11-s − 1.94·13-s + 2.67·14-s + 5/4·16-s + 1.94·17-s − 1.14·19-s + 1.34·20-s + 0.852·22-s + 0.834·23-s + 25-s + 2.74·26-s − 2.83·28-s + 1.48·29-s − 1.06·32-s − 2.74·34-s − 1.69·35-s + 2.30·37-s + 1.62·38-s − 1.26·40-s + 1.56·41-s − 1.06·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.035374870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.035374870\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 7 T + 6 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 8 T - 7 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 9 T + 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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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
−9.585428687017966301534689919044, −9.364644512825325730609100510033, −8.920211559187548006952372772769, −8.539712354737196295293304838421, −7.909829985815034056251231087842, −7.69466666828269671587790405258, −7.07751923120816537298270946943, −7.04696100958362059159871998611, −6.36363726447911907901599170113, −6.16582367212964463769411372808, −5.63128092366337050712361123398, −5.33552855292104117083427593071, −4.64308039796476169767953123550, −4.10866635980823788275476869619, −3.13505736462860934789839399382, −3.01572733333613690952063168664, −2.36123097735642428639897425592, −2.29251804134914533034824105087, −0.924098159746383725742093631560, −0.63929129443516576922908802115,
0.63929129443516576922908802115, 0.924098159746383725742093631560, 2.29251804134914533034824105087, 2.36123097735642428639897425592, 3.01572733333613690952063168664, 3.13505736462860934789839399382, 4.10866635980823788275476869619, 4.64308039796476169767953123550, 5.33552855292104117083427593071, 5.63128092366337050712361123398, 6.16582367212964463769411372808, 6.36363726447911907901599170113, 7.04696100958362059159871998611, 7.07751923120816537298270946943, 7.69466666828269671587790405258, 7.909829985815034056251231087842, 8.539712354737196295293304838421, 8.920211559187548006952372772769, 9.364644512825325730609100510033, 9.585428687017966301534689919044
