| L(s) = 1 | − 1.37e3·49-s − 2.86e3·81-s − 1.06e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 251-s + ⋯ |
| L(s) = 1 | − 4·49-s − 3.92·81-s − 8·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + 0.000270·239-s + 0.000267·241-s + 0.000251·251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.002484712432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.002484712432\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + p^{3} T^{2} )^{4} \) |
| good | 3 | \( ( 1 + 1430 T^{4} + p^{12} T^{8} )^{2} \) |
| 5 | \( ( 1 - 30602 T^{4} + p^{12} T^{8} )^{2} \) |
| 11 | \( ( 1 + p^{3} T^{2} )^{8} \) |
| 13 | \( ( 1 + 3229270 T^{4} + p^{12} T^{8} )^{2} \) |
| 17 | \( ( 1 - p^{3} T^{2} )^{8} \) |
| 19 | \( ( 1 - 20499050 T^{4} + p^{12} T^{8} )^{2} \) |
| 23 | \( ( 1 + 14870 T^{2} + p^{6} T^{4} )^{4} \) |
| 29 | \( ( 1 - p^{3} T^{2} )^{8} \) |
| 31 | \( ( 1 + p^{3} T^{2} )^{8} \) |
| 37 | \( ( 1 - p^{3} T^{2} )^{8} \) |
| 41 | \( ( 1 - p^{3} T^{2} )^{8} \) |
| 43 | \( ( 1 + p^{3} T^{2} )^{8} \) |
| 47 | \( ( 1 + p^{3} T^{2} )^{8} \) |
| 53 | \( ( 1 - p^{3} T^{2} )^{8} \) |
| 59 | \( ( 1 + 39634956790 T^{4} + p^{12} T^{8} )^{2} \) |
| 61 | \( ( 1 - 94184008490 T^{4} + p^{12} T^{8} )^{2} \) |
| 67 | \( ( 1 + p^{3} T^{2} )^{8} \) |
| 71 | \( ( 1 - 332530 T^{2} + p^{6} T^{4} )^{4} \) |
| 73 | \( ( 1 - p^{3} T^{2} )^{8} \) |
| 79 | \( ( 1 + 236990 T^{2} + p^{6} T^{4} )^{4} \) |
| 83 | \( ( 1 - 394188032810 T^{4} + p^{12} T^{8} )^{2} \) |
| 89 | \( ( 1 - p^{3} T^{2} )^{8} \) |
| 97 | \( ( 1 - p^{3} T^{2} )^{8} \) |
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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
−4.49900678278844552207159975258, −4.18143108733312044633982477740, −4.05198123332578450400853862692, −3.90300188624325097045230018132, −3.85754971609703077291336690216, −3.73153955308495356888342593211, −3.64102792771595495195133913148, −3.46810405803364011790964023216, −3.09623645469720351813551881445, −2.85682971651551136898636581894, −2.82067720724481428615679618346, −2.79033727657368637289231385307, −2.71760139727952264750275582902, −2.56051053096698070992297775103, −2.17406397617417002958569692313, −1.93166446993047953228775585072, −1.82944817041684669790062984956, −1.46688505038562832434565787444, −1.42846294876424550960287439321, −1.41360672582054760436099573569, −1.19947543110005745350994509510, −0.815707718414117838481728814507, −0.53057682973718555153734842281, −0.23767520581380971621820726206, −0.00631968518072539353367997759,
0.00631968518072539353367997759, 0.23767520581380971621820726206, 0.53057682973718555153734842281, 0.815707718414117838481728814507, 1.19947543110005745350994509510, 1.41360672582054760436099573569, 1.42846294876424550960287439321, 1.46688505038562832434565787444, 1.82944817041684669790062984956, 1.93166446993047953228775585072, 2.17406397617417002958569692313, 2.56051053096698070992297775103, 2.71760139727952264750275582902, 2.79033727657368637289231385307, 2.82067720724481428615679618346, 2.85682971651551136898636581894, 3.09623645469720351813551881445, 3.46810405803364011790964023216, 3.64102792771595495195133913148, 3.73153955308495356888342593211, 3.85754971609703077291336690216, 3.90300188624325097045230018132, 4.05198123332578450400853862692, 4.18143108733312044633982477740, 4.49900678278844552207159975258
Plot not available for L-functions of degree greater than 10.
