| L(s) = 1 | − 8·9-s + 24·25-s − 16·29-s − 16·37-s + 4·49-s − 16·53-s + 20·81-s + 48·109-s + 48·113-s + 72·121-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 + ⋯ |
| L(s) = 1 | − 8/3·9-s + 24/5·25-s − 2.97·29-s − 2.63·37-s + 4/7·49-s − 2.19·53-s + 20/9·81-s + 4.59·109-s + 4.51·113-s + 6.54·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.287642537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.287642537\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 3 | \( ( 1 + 4 T^{2} + 14 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 - 12 T^{2} + 78 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 12 T^{2} + 366 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | \( ( 1 + 36 T^{2} + 1038 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 20 T^{2} + 646 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 60 T^{2} + 2310 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 4 T^{2} + 3238 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 100 T^{2} + 5686 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 124 T^{2} + 7750 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 59 | \( ( 1 + 132 T^{2} + 10926 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 236 T^{2} + 21358 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 4 T^{2} - 3818 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 196 T^{2} + 18534 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 132 T^{2} + 8742 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 100 T^{2} + 11782 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 196 T^{2} + 19150 T^{4} + 196 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 324 T^{2} + 41958 T^{4} - 324 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 68 T^{2} + 19462 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−5.61940615199338735197907410939, −5.51196794458349198902874630592, −5.28849559450796918614524019911, −5.01145148544023176931176442344, −4.94196251110733237493944550748, −4.86155165324298046699039662024, −4.75251346266309161044823199214, −4.62584078268731707706650157106, −4.27314821578559974208831146540, −4.23713015329586736407332921148, −3.90825183869261432596270871156, −3.53097386546776849171677605374, −3.38136351588905753384185367880, −3.35360941439293188636996161852, −3.34758402955247178487605076665, −3.07772417019787568662543111147, −2.99836652766771273950472596065, −2.55417827599597349752748000297, −2.49698770232681873445592758006, −2.26758508862851611296465021458, −1.83991243010796363635342658179, −1.72011555056152411310077960065, −1.52214246454495651836127650903, −0.74688557967185636121840484526, −0.51298582969417518798000141339,
0.51298582969417518798000141339, 0.74688557967185636121840484526, 1.52214246454495651836127650903, 1.72011555056152411310077960065, 1.83991243010796363635342658179, 2.26758508862851611296465021458, 2.49698770232681873445592758006, 2.55417827599597349752748000297, 2.99836652766771273950472596065, 3.07772417019787568662543111147, 3.34758402955247178487605076665, 3.35360941439293188636996161852, 3.38136351588905753384185367880, 3.53097386546776849171677605374, 3.90825183869261432596270871156, 4.23713015329586736407332921148, 4.27314821578559974208831146540, 4.62584078268731707706650157106, 4.75251346266309161044823199214, 4.86155165324298046699039662024, 4.94196251110733237493944550748, 5.01145148544023176931176442344, 5.28849559450796918614524019911, 5.51196794458349198902874630592, 5.61940615199338735197907410939
Plot not available for L-functions of degree greater than 10.
