L(s) = 1 | + 16·9-s − 32·25-s + 20·49-s + 124·81-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 512·225-s + 227-s + 229-s + ⋯ |
L(s) = 1 | + 16/3·9-s − 6.39·25-s + 20/7·49-s + 13.7·81-s − 3.63·121-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 − 2.46·169-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 − 34.1·225-s + 0.0663·227-s + 0.0660·229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.625734342\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.625734342\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
good | 3 | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{4} \) |
| 5 | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 23 | \( ( 1 - p T^{2} )^{8} \) |
| 29 | \( ( 1 - p T^{2} )^{8} \) |
| 31 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \) |
| 41 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 104 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−4.79088327549167270224957356193, −4.76638366109868051267427951782, −4.54255745672273668848283164285, −4.38639150391389180712307344272, −4.21610532038637666154304484555, −4.15990911221103853884477304436, −3.98826418511634455127760998951, −3.95771310850442420163057043496, −3.71223921972168417243472232252, −3.70181416077068708550471718865, −3.68078553753299007651912355844, −3.54429975884060915681907309625, −3.26212953349621916363413334122, −2.60907780948850379342388754623, −2.60741658325559565487405188492, −2.45271261588049143874118745241, −2.44566525331869703619509702309, −2.19703364262668787585036280148, −1.80743338757989986319870438555, −1.62992531547686236985356964255, −1.60231692087868306763915758852, −1.42943818177249937183192426260, −1.19565744091903906847901562434, −0.967315265489517001562289388018, −0.19655198445435576392837210199,
0.19655198445435576392837210199, 0.967315265489517001562289388018, 1.19565744091903906847901562434, 1.42943818177249937183192426260, 1.60231692087868306763915758852, 1.62992531547686236985356964255, 1.80743338757989986319870438555, 2.19703364262668787585036280148, 2.44566525331869703619509702309, 2.45271261588049143874118745241, 2.60741658325559565487405188492, 2.60907780948850379342388754623, 3.26212953349621916363413334122, 3.54429975884060915681907309625, 3.68078553753299007651912355844, 3.70181416077068708550471718865, 3.71223921972168417243472232252, 3.95771310850442420163057043496, 3.98826418511634455127760998951, 4.15990911221103853884477304436, 4.21610532038637666154304484555, 4.38639150391389180712307344272, 4.54255745672273668848283164285, 4.76638366109868051267427951782, 4.79088327549167270224957356193
Plot not available for L-functions of degree greater than 10.