| L(s) = 1 | + 16·9-s − 24·11-s + 104·23-s − 4·25-s + 32·29-s − 128·37-s + 40·43-s + 24·53-s + 112·67-s + 224·71-s + 240·79-s + 128·81-s − 384·99-s − 40·107-s + 272·109-s + 416·113-s + 72·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 100·169-s + ⋯ |
| L(s) = 1 | + 16/9·9-s − 2.18·11-s + 4.52·23-s − 0.159·25-s + 1.10·29-s − 3.45·37-s + 0.930·43-s + 0.452·53-s + 1.67·67-s + 3.15·71-s + 3.03·79-s + 1.58·81-s − 3.87·99-s − 0.373·107-s + 2.49·109-s + 3.68·113-s + 0.595·121-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 − 0.591·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.655647778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.655647778\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_4\times C_2$ | \( 1 - 16 T^{2} + 128 T^{4} - 16 p^{4} T^{6} + p^{8} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 + 4 T^{2} + 862 T^{4} + 4 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 12 T + 180 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 100 T^{2} + 59230 T^{4} + 100 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 976 T^{2} + 397248 T^{4} - 976 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 384 T^{2} + 62208 T^{4} - 384 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{4} \) |
| 29 | $D_{4}$ | \( ( 1 - 16 T + 1354 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 2276 T^{2} + 2834758 T^{4} - 2276 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $C_2$ | \( ( 1 + 32 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 5984 T^{2} + 14598784 T^{4} - 5984 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 20 T - 1004 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 2804 T^{2} + 3820678 T^{4} - 2804 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 12 T + 5262 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 9168 T^{2} + 39927456 T^{4} - 9168 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 10940 T^{2} + 56671390 T^{4} - 10940 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 56 T + 6234 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 112 T + 11650 T^{2} - 112 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 16032 T^{2} + 115732416 T^{4} - 16032 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 120 T + 15690 T^{2} - 120 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 2448 T^{2} + 41142720 T^{4} - 2448 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2:C_4$ | \( 1 - 12144 T^{2} + 147978144 T^{4} - 12144 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 2304 T^{2} + 176950848 T^{4} - 2304 p^{4} T^{6} + p^{8} T^{8} \) |
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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
−8.961813733513373035795619077199, −8.828561885146152963729543657080, −8.323568393723276890213069998986, −8.085785309008867838626161884106, −7.71255759769044038899841132869, −7.58545465863362541045655794490, −7.20980818319586824041495954221, −7.02897772393246890743267348289, −6.78309784949025463853402608736, −6.52592263625390033130481793227, −6.37905512916371549335488159926, −5.54668271280280194179701329821, −5.27398957588033681924536562068, −5.18003559838064427291551453562, −4.97312307633122218636755627696, −4.76423131820406562183809838948, −4.29237833004364950898630065549, −3.69475979769732155471716620134, −3.53204559644791686187286442502, −2.93473595459345005741890168372, −2.90492350668109238846686420368, −2.03803440281940460598233225562, −1.98434284168408197993555512538, −0.843972249789111576887553692822, −0.827389256127538443670599308860,
0.827389256127538443670599308860, 0.843972249789111576887553692822, 1.98434284168408197993555512538, 2.03803440281940460598233225562, 2.90492350668109238846686420368, 2.93473595459345005741890168372, 3.53204559644791686187286442502, 3.69475979769732155471716620134, 4.29237833004364950898630065549, 4.76423131820406562183809838948, 4.97312307633122218636755627696, 5.18003559838064427291551453562, 5.27398957588033681924536562068, 5.54668271280280194179701329821, 6.37905512916371549335488159926, 6.52592263625390033130481793227, 6.78309784949025463853402608736, 7.02897772393246890743267348289, 7.20980818319586824041495954221, 7.58545465863362541045655794490, 7.71255759769044038899841132869, 8.085785309008867838626161884106, 8.323568393723276890213069998986, 8.828561885146152963729543657080, 8.961813733513373035795619077199
