| L(s) = 1 | − 4·2-s + 12·4-s + 7·5-s − 15·7-s − 32·8-s − 28·10-s − 15·11-s − 20·13-s + 60·14-s + 80·16-s + 16·17-s + 84·20-s + 60·22-s − 6·23-s + 25·25-s + 80·26-s − 180·28-s + 10·29-s + 93·31-s − 192·32-s − 64·34-s − 105·35-s − 20·37-s − 224·40-s − 50·41-s + 30·43-s − 180·44-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s + 7/5·5-s − 2.14·7-s − 4·8-s − 2.79·10-s − 1.36·11-s − 1.53·13-s + 30/7·14-s + 5·16-s + 0.941·17-s + 21/5·20-s + 2.72·22-s − 0.260·23-s + 25-s + 3.07·26-s − 6.42·28-s + 0.344·29-s + 3·31-s − 6·32-s − 1.88·34-s − 3·35-s − 0.540·37-s − 5.59·40-s − 1.21·41-s + 0.697·43-s − 4.09·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3122958642\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3122958642\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 7 T + 24 T^{2} - 7 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )( 1 + 13 T + p^{2} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 15 T + 196 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 20 T + 231 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 614 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 541 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T - 741 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{2}( 1 - p T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 50 T + 819 T^{2} + 50 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 30 T + 2149 T^{2} - 30 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 150 T + 9709 T^{2} + 150 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 60 T + 4681 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 64 T + 375 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 150 T + 11989 T^{2} + 150 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 55 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 12 T + 6289 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 51 T + 7756 T^{2} + 51 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 25 T - 8784 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.83534237889738561618289451562, −10.60470934529849495584554308914, −10.22774447631534498251451063142, −10.20049488523174416000889476573, −9.684570676276880482042935687614, −9.672932136812196961680077788853, −9.032599436762012152624028636664, −8.208362168258312729420778035074, −8.145784791023361266885044068555, −7.28441907538008430125041175387, −6.85878060497306822595558233077, −6.51685093241890692334457621816, −5.95034344436384526740719132368, −5.59068101142978692765560267028, −4.81812797296731777702204904737, −3.23375591722695956839539515490, −2.82353594665142998283137534212, −2.56864556666291773799553869047, −1.55048453603780783945271853919, −0.34851256136412525315155256919,
0.34851256136412525315155256919, 1.55048453603780783945271853919, 2.56864556666291773799553869047, 2.82353594665142998283137534212, 3.23375591722695956839539515490, 4.81812797296731777702204904737, 5.59068101142978692765560267028, 5.95034344436384526740719132368, 6.51685093241890692334457621816, 6.85878060497306822595558233077, 7.28441907538008430125041175387, 8.145784791023361266885044068555, 8.208362168258312729420778035074, 9.032599436762012152624028636664, 9.672932136812196961680077788853, 9.684570676276880482042935687614, 10.20049488523174416000889476573, 10.22774447631534498251451063142, 10.60470934529849495584554308914, 11.83534237889738561618289451562
