| L(s) = 1 | + 6·3-s + 3·5-s + 2·7-s + 27·9-s + 15·11-s − 23·13-s + 18·15-s − 34·17-s + 137·19-s + 12·21-s + 201·23-s − 193·25-s + 108·27-s − 66·29-s + 464·31-s + 90·33-s + 6·35-s − 140·37-s − 138·39-s − 45·41-s + 407·43-s + 81·45-s + 114·47-s − 482·49-s − 204·51-s + 792·53-s + 45·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.268·5-s + 0.107·7-s + 9-s + 0.411·11-s − 0.490·13-s + 0.309·15-s − 0.485·17-s + 1.65·19-s + 0.124·21-s + 1.82·23-s − 1.54·25-s + 0.769·27-s − 0.422·29-s + 2.68·31-s + 0.474·33-s + 0.0289·35-s − 0.622·37-s − 0.566·39-s − 0.171·41-s + 1.44·43-s + 0.268·45-s + 0.353·47-s − 1.40·49-s − 0.560·51-s + 2.05·53-s + 0.110·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 665856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 665856 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.607508917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.607508917\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 3 T + 202 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 486 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 15 T + 1462 T^{2} - 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 23 T + 2064 T^{2} + 23 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 137 T + 17154 T^{2} - 137 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 201 T + 34384 T^{2} - 201 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 66 T + 25546 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 464 T + 106170 T^{2} - 464 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 140 T - 29670 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 45 T + 36592 T^{2} + 45 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 407 T + 178266 T^{2} - 407 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 114 T + 85270 T^{2} - 114 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 792 T + 357286 T^{2} - 792 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 570 T + 467662 T^{2} + 570 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 16 T + 248202 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 208 T + 454758 T^{2} + 208 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 90 T + 716038 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 p T + p^{3} T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 764 T + 1103058 T^{2} - 764 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 138 T + 902110 T^{2} - 138 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 654 T + 768946 T^{2} - 654 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 146 T + 1082754 T^{2} + 146 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−9.850418641384548696527887464491, −9.597714813829746035281433612415, −9.074486400134581859404249463038, −9.074216963785299372071046174557, −8.295058783412092579409037897190, −8.006734036988850856298238972935, −7.44148831419494020915863749289, −7.30313087754014174763733251713, −6.55763512103065883410628321829, −6.37822350659271709435643637579, −5.45281473529071606060254848236, −5.27631944498929056491434767062, −4.39591696797228239335789770896, −4.34344323047402360931541384687, −3.36874161883012455346597017295, −3.14936149689019291169557141968, −2.50595583320225255017454040756, −1.99944146754355604165535378692, −1.20745149444770205506083954523, −0.69356551879258903859245915809,
0.69356551879258903859245915809, 1.20745149444770205506083954523, 1.99944146754355604165535378692, 2.50595583320225255017454040756, 3.14936149689019291169557141968, 3.36874161883012455346597017295, 4.34344323047402360931541384687, 4.39591696797228239335789770896, 5.27631944498929056491434767062, 5.45281473529071606060254848236, 6.37822350659271709435643637579, 6.55763512103065883410628321829, 7.30313087754014174763733251713, 7.44148831419494020915863749289, 8.006734036988850856298238972935, 8.295058783412092579409037897190, 9.074216963785299372071046174557, 9.074486400134581859404249463038, 9.597714813829746035281433612415, 9.850418641384548696527887464491
