| L(s) = 1 | − 6·3-s + 34·7-s + 27·9-s − 20·19-s − 204·21-s + 244·25-s − 108·27-s − 252·29-s + 16·31-s − 488·37-s − 360·47-s + 813·49-s − 1.18e3·53-s + 120·57-s + 1.08e3·59-s + 918·63-s − 1.46e3·75-s + 405·81-s + 1.72e3·83-s + 1.51e3·87-s − 96·93-s + 2.75e3·103-s − 1.20e3·109-s + 2.92e3·111-s − 4.21e3·113-s − 512·121-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.83·7-s + 9-s − 0.241·19-s − 2.11·21-s + 1.95·25-s − 0.769·27-s − 1.61·29-s + 0.0926·31-s − 2.16·37-s − 1.11·47-s + 2.37·49-s − 3.07·53-s + 0.278·57-s + 2.38·59-s + 1.83·63-s − 2.25·75-s + 5/9·81-s + 2.28·83-s + 1.86·87-s − 0.107·93-s + 2.63·103-s − 1.05·109-s + 2.50·111-s − 3.50·113-s − 0.384·121-s + 0.000698·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.525351879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.525351879\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | $C_2$ | \( 1 - 34 T + p^{3} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 244 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 512 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 1006 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7180 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 10 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 11080 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 126 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 244 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 1036 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 132878 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 180 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 594 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 540 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 329546 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 518218 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 683848 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 340634 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 72322 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 864 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 256436 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1171946 T^{2} + p^{6} T^{4} \) |
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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
−9.448001099339986627092986358136, −8.975165549972292657009056436943, −8.634335683192845940198397297930, −8.169784401231237289697586373786, −7.86195173078038100511859199845, −7.25711406413058172140102811606, −7.12349798831823867290529638516, −6.35742102097622470171720506237, −6.34970762307558738207276777907, −5.44357631328977871172550891606, −5.20906194549851680531876535384, −4.87910886545696546158312362944, −4.69234870840719239523814357860, −3.81262038954556358874276243359, −3.62237462880617613807716818055, −2.71523567805671672970698901090, −1.99995072599308147719431826438, −1.56642233716957359547819770753, −1.13189422119766007142778903966, −0.33036392804732700552314370165,
0.33036392804732700552314370165, 1.13189422119766007142778903966, 1.56642233716957359547819770753, 1.99995072599308147719431826438, 2.71523567805671672970698901090, 3.62237462880617613807716818055, 3.81262038954556358874276243359, 4.69234870840719239523814357860, 4.87910886545696546158312362944, 5.20906194549851680531876535384, 5.44357631328977871172550891606, 6.34970762307558738207276777907, 6.35742102097622470171720506237, 7.12349798831823867290529638516, 7.25711406413058172140102811606, 7.86195173078038100511859199845, 8.169784401231237289697586373786, 8.634335683192845940198397297930, 8.975165549972292657009056436943, 9.448001099339986627092986358136
