| L(s) = 1 | + 4·13-s − 4·25-s − 32·43-s + 38·49-s − 4·61-s − 4·79-s − 8·103-s + 46·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 24·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 1.10·13-s − 4/5·25-s − 4.87·43-s + 38/7·49-s − 0.512·61-s − 0.450·79-s − 0.788·103-s + 4.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8097590492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8097590492\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 + T^{2} )^{4} \) |
| 13 | \( ( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 7 | \( ( 1 - 19 T^{2} + 180 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 23 T^{2} + 300 T^{4} - 23 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 23 T^{2} + 636 T^{4} + 23 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 28 T^{2} + 390 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + p T^{2} + 192 T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 80 T^{2} + 3150 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 76 T^{2} + 2838 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 103 T^{2} + 5316 T^{4} - 103 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 143 T^{2} + 8400 T^{4} - 143 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 47 | \( ( 1 - 104 T^{2} + 5934 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 119 T^{2} + 7764 T^{4} + 119 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 152 T^{2} + 11550 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + T + 48 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 - 208 T^{2} + 19662 T^{4} - 208 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 95 T^{2} + 6324 T^{4} - 95 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 112 T^{2} + 12606 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + T + 150 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - 335 T^{2} + 43824 T^{4} - 335 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 319 T^{2} + 43260 T^{4} - 319 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−3.76270885295704989141315879500, −3.63919204604444100624629792198, −3.63113172476790926470759695099, −3.32756748076090941448603042423, −3.21684435202438720353012534680, −3.13347438714931250870554881824, −3.11752802338585207760200721471, −3.11340693008362479910895315528, −2.76242313108703904084057453886, −2.45254061367822492155645929075, −2.42406808406277003553926994635, −2.41419407691118934172803139392, −2.39666905727654047020594597739, −2.03521641045187954345658986660, −1.94607404973274462093432784408, −1.88785475018526271783213974427, −1.60989701956771739664543260150, −1.49217457678181642095760308247, −1.42496665889788083395878068545, −1.20459953067396699200292574637, −1.05601067177133197665713024846, −0.68563135126300476563900286382, −0.61761876443663113048670777010, −0.53803223242451943068828879157, −0.06969665510498629719788778905,
0.06969665510498629719788778905, 0.53803223242451943068828879157, 0.61761876443663113048670777010, 0.68563135126300476563900286382, 1.05601067177133197665713024846, 1.20459953067396699200292574637, 1.42496665889788083395878068545, 1.49217457678181642095760308247, 1.60989701956771739664543260150, 1.88785475018526271783213974427, 1.94607404973274462093432784408, 2.03521641045187954345658986660, 2.39666905727654047020594597739, 2.41419407691118934172803139392, 2.42406808406277003553926994635, 2.45254061367822492155645929075, 2.76242313108703904084057453886, 3.11340693008362479910895315528, 3.11752802338585207760200721471, 3.13347438714931250870554881824, 3.21684435202438720353012534680, 3.32756748076090941448603042423, 3.63113172476790926470759695099, 3.63919204604444100624629792198, 3.76270885295704989141315879500
Plot not available for L-functions of degree greater than 10.
