| L(s) = 1 | + 5·9-s − 4·19-s + 6·29-s + 10·31-s + 6·41-s + 10·49-s + 24·59-s + 28·61-s − 30·71-s + 20·79-s + 16·81-s + 12·101-s + 32·109-s − 22·121-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 + 179-s + ⋯ |
| L(s) = 1 | + 5/3·9-s − 0.917·19-s + 1.11·29-s + 1.79·31-s + 0.937·41-s + 10/7·49-s + 3.12·59-s + 3.58·61-s − 3.56·71-s + 2.25·79-s + 16/9·81-s + 1.19·101-s + 3.06·109-s − 2·121-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 + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.645254468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.645254468\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.087116303443760518142614216280, −8.831456210107175007266278866168, −8.314211003121809836656248374167, −8.235206556170506111986937114080, −7.55898345363030104854157684122, −7.24712386481719225679223675556, −6.78419227594162703006681007757, −6.73709704122140940156918410691, −5.97357821525095593476757098175, −5.89607936875442353457127825616, −4.98323577386822021303552664044, −4.91276884538335403575254288265, −4.19723181045914320307516262559, −4.12769631603432109831009053895, −3.61545645281537600779155628910, −2.90009683331740490879665127060, −2.23772419551646112198560166838, −2.12199275246863832848412938979, −0.968204168022438439367862263148, −0.882161353392458218712659694966,
0.882161353392458218712659694966, 0.968204168022438439367862263148, 2.12199275246863832848412938979, 2.23772419551646112198560166838, 2.90009683331740490879665127060, 3.61545645281537600779155628910, 4.12769631603432109831009053895, 4.19723181045914320307516262559, 4.91276884538335403575254288265, 4.98323577386822021303552664044, 5.89607936875442353457127825616, 5.97357821525095593476757098175, 6.73709704122140940156918410691, 6.78419227594162703006681007757, 7.24712386481719225679223675556, 7.55898345363030104854157684122, 8.235206556170506111986937114080, 8.314211003121809836656248374167, 8.831456210107175007266278866168, 9.087116303443760518142614216280
