L(s) = 1 | + 2·25-s − 2·81-s − 12·89-s + 12·101-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-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 | + 2·25-s − 2·81-s − 12·89-s + 12·101-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.102743027\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.102743027\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{4} )^{2} \) |
| 5 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 7 | \( 1 - T^{4} + T^{8} \) |
good | 11 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 13 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 17 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 19 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 23 | \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 29 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 31 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 37 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 41 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 43 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 47 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 53 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 59 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 61 | \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | \( ( 1 + T^{4} )^{2}( 1 - T^{4} + T^{8} ) \) |
| 71 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 73 | \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \) |
| 79 | \( ( 1 - T^{2} + T^{4} )^{4} \) |
| 83 | \( ( 1 - T^{4} + T^{8} )^{2} \) |
| 89 | \( ( 1 + T )^{8}( 1 + T + T^{2} )^{4} \) |
| 97 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.77116944340870452643935491305, −3.73616002315054518555519974145, −3.50958670979317578234957647762, −3.47535536259215937973744419663, −3.26094377914878128619017677329, −3.04911743384282214052360479200, −3.03193268549033210467863904678, −3.02664281931911211285422845476, −2.84223756950420830526606167491, −2.75654204861734006317750615036, −2.67187324296023750416213216565, −2.57072517571473936731330734536, −2.44449803557646173338612020518, −2.20379385791908296630401422755, −2.05588211203471848073006824844, −1.85402857033447995527481432060, −1.71639902272188519168533392137, −1.64990733825982099400825637658, −1.56593148908373671913782018166, −1.56159748332681379007208729406, −1.18001691211894767998804880168, −1.00817362582173333841271347532, −0.827950920659224598987851089116, −0.66090180338384530802245674140, −0.42669062067821938387986788612,
0.42669062067821938387986788612, 0.66090180338384530802245674140, 0.827950920659224598987851089116, 1.00817362582173333841271347532, 1.18001691211894767998804880168, 1.56159748332681379007208729406, 1.56593148908373671913782018166, 1.64990733825982099400825637658, 1.71639902272188519168533392137, 1.85402857033447995527481432060, 2.05588211203471848073006824844, 2.20379385791908296630401422755, 2.44449803557646173338612020518, 2.57072517571473936731330734536, 2.67187324296023750416213216565, 2.75654204861734006317750615036, 2.84223756950420830526606167491, 3.02664281931911211285422845476, 3.03193268549033210467863904678, 3.04911743384282214052360479200, 3.26094377914878128619017677329, 3.47535536259215937973744419663, 3.50958670979317578234957647762, 3.73616002315054518555519974145, 3.77116944340870452643935491305
Plot not available for L-functions of degree greater than 10.