| L(s) = 1 | − 4·11-s + 16-s + 4·71-s − 2·81-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 4·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 4·11-s + 16-s + 4·71-s − 2·81-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 4·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5375832316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5375832316\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 3 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
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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
−7.12911639013463136000110295451, −6.95859991922082585241453399050, −6.91240748978798720574315366460, −6.59709413734572691190127769182, −6.06864411573423634846182634193, −5.89244075884852514837005178803, −5.84404067243340465214747941895, −5.70432737776668382567047966735, −5.20173850773963351762683898968, −5.15318730674635754156384285009, −5.14265728771490066099987508288, −4.68077485908352453628953452659, −4.60297917776591991362979888643, −4.28739158381938449392032060958, −3.87326402879235251066294548054, −3.41331898879863638145969759964, −3.39516802043105116080282867516, −3.33972925965622599452415660909, −2.64888801856488417467285598299, −2.49933563134259233916872440427, −2.37978861687901926380447512796, −2.28612800923728337838285302019, −1.53842539097858420689547349060, −1.26785764692904594031077648994, −0.50897020455499073638655761821,
0.50897020455499073638655761821, 1.26785764692904594031077648994, 1.53842539097858420689547349060, 2.28612800923728337838285302019, 2.37978861687901926380447512796, 2.49933563134259233916872440427, 2.64888801856488417467285598299, 3.33972925965622599452415660909, 3.39516802043105116080282867516, 3.41331898879863638145969759964, 3.87326402879235251066294548054, 4.28739158381938449392032060958, 4.60297917776591991362979888643, 4.68077485908352453628953452659, 5.14265728771490066099987508288, 5.15318730674635754156384285009, 5.20173850773963351762683898968, 5.70432737776668382567047966735, 5.84404067243340465214747941895, 5.89244075884852514837005178803, 6.06864411573423634846182634193, 6.59709413734572691190127769182, 6.91240748978798720574315366460, 6.95859991922082585241453399050, 7.12911639013463136000110295451
