| L(s) = 1 | + 7-s + 3·19-s − 25-s − 3·31-s + 2·37-s + 2·43-s − 3·61-s − 2·67-s − 3·73-s − 2·79-s − 109-s + 121-s + 127-s + 131-s + 3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s − 175-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 7-s + 3·19-s − 25-s − 3·31-s + 2·37-s + 2·43-s − 3·61-s − 2·67-s − 3·73-s − 2·79-s − 109-s + 121-s + 127-s + 131-s + 3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s − 175-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.035121402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.035121402\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−10.68102768420910412258500959013, −10.52763465050665019733674135406, −9.640117839765748790219414294167, −9.609291742336016477735603166610, −8.993744850455697861459664885527, −8.897696436817363988728595590791, −7.981470997200612890847591784955, −7.57121436215882822954182434071, −7.39122724278161595238615136385, −7.32121848030643215348758627346, −6.04622910470620979602235280797, −5.95869670364095433598890288696, −5.41675416550233053352327716090, −5.04931520994212375136737243929, −4.22198973540775874696625740554, −4.10737781036924716793337064531, −2.99628300238794761710597513424, −2.96844602323825526379510974099, −1.72959302221823731376959918275, −1.34901770569908910157720622206,
1.34901770569908910157720622206, 1.72959302221823731376959918275, 2.96844602323825526379510974099, 2.99628300238794761710597513424, 4.10737781036924716793337064531, 4.22198973540775874696625740554, 5.04931520994212375136737243929, 5.41675416550233053352327716090, 5.95869670364095433598890288696, 6.04622910470620979602235280797, 7.32121848030643215348758627346, 7.39122724278161595238615136385, 7.57121436215882822954182434071, 7.981470997200612890847591784955, 8.897696436817363988728595590791, 8.993744850455697861459664885527, 9.609291742336016477735603166610, 9.640117839765748790219414294167, 10.52763465050665019733674135406, 10.68102768420910412258500959013
