| L(s) = 1 | + 5·9-s − 6·11-s + 4·19-s + 18·29-s − 16·31-s − 49-s + 24·59-s + 16·61-s + 10·79-s + 16·81-s − 24·89-s − 30·99-s − 12·101-s + 38·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + 20·171-s + 173-s + ⋯ |
| L(s) = 1 | + 5/3·9-s − 1.80·11-s + 0.917·19-s + 3.34·29-s − 2.87·31-s − 1/7·49-s + 3.12·59-s + 2.04·61-s + 1.12·79-s + 16/9·81-s − 2.54·89-s − 3.01·99-s − 1.19·101-s + 3.63·109-s + 5/11·121-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 + 1.92·169-s + 1.52·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.891434618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.891434618\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.880631871836227922586445338244, −8.574901904478214231754256137834, −8.155540271893535838644369900654, −7.929934380071319571904516191256, −7.28673438779604457400750740585, −7.25245733959553561547882982114, −6.75029976839413447489882238920, −6.58497386198627546347670333010, −5.71338860406484897796926891192, −5.45425546604032510273831535691, −5.15594276701776437639526348154, −4.78544845855220300175518105108, −4.09308453566724708132214617670, −4.07567600208962810040399844629, −3.12436376633970318869610686510, −3.08573142890458198990343171127, −2.10470774340178935510859443156, −2.09695979403773195524816727514, −1.08554577552310083870792997118, −0.63231552227700797838597392081,
0.63231552227700797838597392081, 1.08554577552310083870792997118, 2.09695979403773195524816727514, 2.10470774340178935510859443156, 3.08573142890458198990343171127, 3.12436376633970318869610686510, 4.07567600208962810040399844629, 4.09308453566724708132214617670, 4.78544845855220300175518105108, 5.15594276701776437639526348154, 5.45425546604032510273831535691, 5.71338860406484897796926891192, 6.58497386198627546347670333010, 6.75029976839413447489882238920, 7.25245733959553561547882982114, 7.28673438779604457400750740585, 7.929934380071319571904516191256, 8.155540271893535838644369900654, 8.574901904478214231754256137834, 8.880631871836227922586445338244
