| L(s) = 1 | − 9-s + 4·23-s − 25-s + 4·47-s + 2·49-s + 81-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 4·207-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 9-s + 4·23-s − 25-s + 4·47-s + 2·49-s + 81-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 4·207-s + 211-s + 223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.554184036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.554184036\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_1$ | \( ( 1 - T )^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−8.815605378150077745176798134046, −8.803172252870795808587087045302, −8.131882253346117874243983537085, −7.65942888837444445613351455006, −7.35225521314762163027192921704, −7.01754204371703465944853018573, −6.84532313100644106640949245329, −6.17419229198896912771458489542, −5.73939792958227502205846794716, −5.58571706957251820920559843344, −5.19438549741416169212925607599, −4.71629996617010633646601937752, −4.32319360571361173591249317505, −3.82739626350198205000857293738, −3.31673430582043928500690988409, −2.99518096199693647980945657087, −2.35909283230605317562624558027, −2.31274370408054287204785803952, −1.12127492659843845001839173917, −0.861930269953059335073922232095,
0.861930269953059335073922232095, 1.12127492659843845001839173917, 2.31274370408054287204785803952, 2.35909283230605317562624558027, 2.99518096199693647980945657087, 3.31673430582043928500690988409, 3.82739626350198205000857293738, 4.32319360571361173591249317505, 4.71629996617010633646601937752, 5.19438549741416169212925607599, 5.58571706957251820920559843344, 5.73939792958227502205846794716, 6.17419229198896912771458489542, 6.84532313100644106640949245329, 7.01754204371703465944853018573, 7.35225521314762163027192921704, 7.65942888837444445613351455006, 8.131882253346117874243983537085, 8.803172252870795808587087045302, 8.815605378150077745176798134046
