L(s) = 1 | + 8·7-s − 4·13-s + 16·19-s − 8·31-s + 20·37-s − 16·43-s + 34·49-s + 28·61-s + 32·67-s + 20·73-s − 8·79-s − 32·91-s − 28·97-s − 40·103-s + 4·109-s − 22·121-s + 127-s + 131-s + 128·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + ⋯ |
L(s) = 1 | + 3.02·7-s − 1.10·13-s + 3.67·19-s − 1.43·31-s + 3.28·37-s − 2.43·43-s + 34/7·49-s + 3.58·61-s + 3.90·67-s + 2.34·73-s − 0.900·79-s − 3.35·91-s − 2.84·97-s − 3.94·103-s + 0.383·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 11.0·133-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 − 1.07·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.539006381\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.539006381\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−12.35280846304710844017859785578, −11.48744893569626099823233253324, −11.48744893569626099823233253324, −11.20255566161945013765549932153, −11.20255566161945013765549932153, −10.02595561869790412734353017930, −10.02595561869790412734353017930, −9.351967908821189871222500143687, −9.351967908821189871222500143687, −8.239034030776179718908314632866, −8.239034030776179718908314632866, −7.71900234701090149632840343207, −7.71900234701090149632840343207, −6.84353614161574326267996849514, −6.84353614161574326267996849514, −5.45300314870917608103230012398, −5.45300314870917608103230012398, −4.98062769154464971467686882983, −4.98062769154464971467686882983, −3.83217595589008344467032618819, −3.83217595589008344467032618819, −2.47608687475654661435538320080, −2.47608687475654661435538320080, −1.22304458410378265801798735343, −1.22304458410378265801798735343,
1.22304458410378265801798735343, 1.22304458410378265801798735343, 2.47608687475654661435538320080, 2.47608687475654661435538320080, 3.83217595589008344467032618819, 3.83217595589008344467032618819, 4.98062769154464971467686882983, 4.98062769154464971467686882983, 5.45300314870917608103230012398, 5.45300314870917608103230012398, 6.84353614161574326267996849514, 6.84353614161574326267996849514, 7.71900234701090149632840343207, 7.71900234701090149632840343207, 8.239034030776179718908314632866, 8.239034030776179718908314632866, 9.351967908821189871222500143687, 9.351967908821189871222500143687, 10.02595561869790412734353017930, 10.02595561869790412734353017930, 11.20255566161945013765549932153, 11.20255566161945013765549932153, 11.48744893569626099823233253324, 11.48744893569626099823233253324, 12.35280846304710844017859785578