| L(s) = 1 | + 24·7-s + 14·9-s − 36·25-s − 88·31-s + 164·49-s + 336·63-s − 328·73-s + 40·79-s + 115·81-s − 376·97-s + 536·103-s − 420·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 476·169-s + 173-s − 864·175-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 24/7·7-s + 14/9·9-s − 1.43·25-s − 2.83·31-s + 3.34·49-s + 16/3·63-s − 4.49·73-s + 0.506·79-s + 1.41·81-s − 3.87·97-s + 5.20·103-s − 3.47·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.81·169-s + 0.00578·173-s − 4.93·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.026523633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.026523633\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - 14 T^{2} + p^{4} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + 210 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2}( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 718 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 930 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 1394 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 2702 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2210 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 3026 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 190 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 1746 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 1554 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 8974 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 5406 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 82 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 8370 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 14690 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 94 T + p^{2} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.25452655274907011559472504234, −7.22380535663291817756595380499, −6.86159388699743650356630626279, −6.52636106364771798566693373061, −6.30079371806039156889915468798, −5.84111864986540933802151906691, −5.75376409033570148714038904854, −5.57512184067441829725147637285, −5.21464189783455695377190238288, −5.01491595816041265019629702989, −4.74996645028537781602980107507, −4.54670303268171059139487069080, −4.38962434286882916640837104821, −4.25620396100775565137892687840, −3.66347307191241547862920928506, −3.65756871604926836701494980801, −3.38101891999286757384316797068, −2.79983367555593361104335153374, −2.32179501457604473347131766545, −2.12188415614471564829359487149, −1.73923210825111024281738051487, −1.56810362087848676518952423756, −1.37351905895664803529774411579, −1.11745288644848325512040077586, −0.12816175011364755588063338365,
0.12816175011364755588063338365, 1.11745288644848325512040077586, 1.37351905895664803529774411579, 1.56810362087848676518952423756, 1.73923210825111024281738051487, 2.12188415614471564829359487149, 2.32179501457604473347131766545, 2.79983367555593361104335153374, 3.38101891999286757384316797068, 3.65756871604926836701494980801, 3.66347307191241547862920928506, 4.25620396100775565137892687840, 4.38962434286882916640837104821, 4.54670303268171059139487069080, 4.74996645028537781602980107507, 5.01491595816041265019629702989, 5.21464189783455695377190238288, 5.57512184067441829725147637285, 5.75376409033570148714038904854, 5.84111864986540933802151906691, 6.30079371806039156889915468798, 6.52636106364771798566693373061, 6.86159388699743650356630626279, 7.22380535663291817756595380499, 7.25452655274907011559472504234
