| L(s) = 1 | − 4·4-s + 10·16-s − 20·64-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 + 239-s + 241-s + ⋯ |
| L(s) = 1 | − 4·4-s + 10·16-s − 20·64-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 + 239-s + 241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2844419359\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2844419359\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{4} \) |
| 17 | \( 1 \) |
| good | 3 | \( ( 1 + T^{8} )^{2} \) |
| 5 | \( ( 1 + T^{4} )^{4} \) |
| 7 | \( ( 1 + T^{4} )^{4} \) |
| 11 | \( ( 1 + T^{8} )^{2} \) |
| 13 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 19 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 23 | \( ( 1 + T^{4} )^{4} \) |
| 29 | \( ( 1 + T^{4} )^{4} \) |
| 31 | \( ( 1 + T^{4} )^{4} \) |
| 37 | \( ( 1 + T^{4} )^{4} \) |
| 41 | \( ( 1 + T^{8} )^{2} \) |
| 43 | \( ( 1 + T^{4} )^{4} \) |
| 47 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 53 | \( ( 1 + T^{2} )^{8} \) |
| 59 | \( ( 1 + T^{4} )^{4} \) |
| 61 | \( ( 1 + T^{4} )^{4} \) |
| 67 | \( ( 1 + T^{4} )^{4} \) |
| 71 | \( ( 1 + T^{4} )^{4} \) |
| 73 | \( ( 1 + T^{8} )^{2} \) |
| 79 | \( ( 1 + T^{4} )^{4} \) |
| 83 | \( ( 1 + T^{4} )^{4} \) |
| 89 | \( ( 1 + T^{4} )^{4} \) |
| 97 | \( ( 1 + T^{8} )^{2} \) |
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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
−3.91041145608858048876225568197, −3.88812081554661159980427203610, −3.85945412540991370234888401435, −3.77631790482260275895402657340, −3.54367551883035712019517088604, −3.44866184685098969778769136389, −3.35807395878694034466747875955, −3.07635294742206398525582105559, −3.02234452380382829254359401375, −2.99174891031813312693217835031, −2.97468205553403247791573493656, −2.76874431148807665957516071096, −2.70287357465622842631434955215, −2.13356710857894642333339238677, −2.13315064029671857441814791339, −2.05945636248907190596913120402, −2.03474123137513692719430069722, −1.82297219849157610232792367830, −1.44934770163841037696702416150, −1.37550194645472973277388915930, −1.09959386868838736350863992907, −1.09607531357877012942973868086, −0.927467098985324896442934865830, −0.61474123050674484677226533537, −0.31573344612786096436728601680,
0.31573344612786096436728601680, 0.61474123050674484677226533537, 0.927467098985324896442934865830, 1.09607531357877012942973868086, 1.09959386868838736350863992907, 1.37550194645472973277388915930, 1.44934770163841037696702416150, 1.82297219849157610232792367830, 2.03474123137513692719430069722, 2.05945636248907190596913120402, 2.13315064029671857441814791339, 2.13356710857894642333339238677, 2.70287357465622842631434955215, 2.76874431148807665957516071096, 2.97468205553403247791573493656, 2.99174891031813312693217835031, 3.02234452380382829254359401375, 3.07635294742206398525582105559, 3.35807395878694034466747875955, 3.44866184685098969778769136389, 3.54367551883035712019517088604, 3.77631790482260275895402657340, 3.85945412540991370234888401435, 3.88812081554661159980427203610, 3.91041145608858048876225568197
Plot not available for L-functions of degree greater than 10.
