| L(s) = 1 | − 4·2-s + 9·3-s + 66·5-s − 36·6-s + 64·8-s − 264·10-s + 60·11-s − 1.31e3·13-s + 594·15-s − 256·16-s + 414·17-s − 956·19-s − 240·22-s − 600·23-s + 576·24-s + 3.12e3·25-s + 5.26e3·26-s − 729·27-s + 1.11e4·29-s − 2.37e3·30-s + 3.59e3·31-s + 540·33-s − 1.65e3·34-s + 8.45e3·37-s + 3.82e3·38-s − 1.18e4·39-s + 4.22e3·40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1.18·5-s − 0.408·6-s + 0.353·8-s − 0.834·10-s + 0.149·11-s − 2.15·13-s + 0.681·15-s − 1/4·16-s + 0.347·17-s − 0.607·19-s − 0.105·22-s − 0.236·23-s + 0.204·24-s + 25-s + 1.52·26-s − 0.192·27-s + 2.46·29-s − 0.481·30-s + 0.671·31-s + 0.0863·33-s − 0.245·34-s + 1.01·37-s + 0.429·38-s − 1.24·39-s + 0.417·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.721073084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.721073084\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T + p^{4} T^{2} \) |
| 3 | $C_2$ | \( 1 - p^{2} T + p^{4} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 66 T + 1231 T^{2} - 66 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 60 T - 157451 T^{2} - 60 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 658 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 414 T - 1248461 T^{2} - 414 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 956 T - 1562163 T^{2} + 956 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 600 T - 6076343 T^{2} + 600 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5574 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 3592 T - 15726687 T^{2} - 3592 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 8458 T + 2193807 T^{2} - 8458 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 19194 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 13316 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 19680 T + 157957393 T^{2} - 19680 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 31266 T + 559367263 T^{2} - 31266 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26340 T - 21128699 T^{2} + 26340 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 31090 T + 121991799 T^{2} - 31090 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 16804 T - 1067750691 T^{2} - 16804 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6120 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 25558 T - 1419860229 T^{2} - 25558 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 74408 T + 2459494065 T^{2} + 74408 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6468 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 32742 T - 4512020885 T^{2} - 32742 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 166082 T + p^{5} T^{2} )^{2} \) |
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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
−10.70751959053974537890711761330, −10.70164319644372487192447447604, −9.931922611168136564484775835678, −9.829114770807773502981612412812, −9.182239136121496288372983356757, −9.087613752775398076559848473026, −8.345583650632216559569705467367, −7.88039616301310903751840366734, −7.35466287442912369450250878399, −7.01913305727998631326584564022, −6.04566427734473286658921363417, −5.98789510655876439136376803698, −5.10197456290328277513606892993, −4.46568161348973520931320141865, −4.17554912735542614155739267163, −2.84261415547512086719710531418, −2.36407344547398290466223044517, −2.33572313192927834727374480225, −0.890096801724721471513467442261, −0.76277420677127378551363501070,
0.76277420677127378551363501070, 0.890096801724721471513467442261, 2.33572313192927834727374480225, 2.36407344547398290466223044517, 2.84261415547512086719710531418, 4.17554912735542614155739267163, 4.46568161348973520931320141865, 5.10197456290328277513606892993, 5.98789510655876439136376803698, 6.04566427734473286658921363417, 7.01913305727998631326584564022, 7.35466287442912369450250878399, 7.88039616301310903751840366734, 8.345583650632216559569705467367, 9.087613752775398076559848473026, 9.182239136121496288372983356757, 9.829114770807773502981612412812, 9.931922611168136564484775835678, 10.70164319644372487192447447604, 10.70751959053974537890711761330
