| L(s) = 1 | + 2·5-s + 2·11-s − 8·19-s − 25-s + 4·29-s + 16·31-s + 12·41-s + 10·49-s + 4·55-s + 8·59-s + 28·61-s − 16·71-s + 16·79-s − 20·89-s − 16·95-s + 20·101-s + 28·109-s + 3·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s + 32·155-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.603·11-s − 1.83·19-s − 1/5·25-s + 0.742·29-s + 2.87·31-s + 1.87·41-s + 10/7·49-s + 0.539·55-s + 1.04·59-s + 3.58·61-s − 1.89·71-s + 1.80·79-s − 2.11·89-s − 1.64·95-s + 1.99·101-s + 2.68·109-s + 3/11·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + 0.0813·151-s + 2.57·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15681600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.262624915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.262624915\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.561690076916054853886704956183, −8.509151204991037720525387181518, −7.969278943331290246984175117134, −7.56380120870672226378041199914, −7.07471794438814770083035953668, −6.68899086485885057007120434305, −6.39804456224879840857164182770, −6.17149480136203216358665343108, −5.66882303551193056114112541739, −5.48925641320362647852716259597, −4.67497077699503002441332596121, −4.58221927067700631649930411133, −4.02440603694632319159604924134, −3.86762120371455746166277772484, −3.00571192406293325630737643907, −2.65791307822691878667918729706, −2.15097134920176642783220814059, −1.95780733126578720706026041785, −0.913959038410848158142972276161, −0.76682447496435372864109877362,
0.76682447496435372864109877362, 0.913959038410848158142972276161, 1.95780733126578720706026041785, 2.15097134920176642783220814059, 2.65791307822691878667918729706, 3.00571192406293325630737643907, 3.86762120371455746166277772484, 4.02440603694632319159604924134, 4.58221927067700631649930411133, 4.67497077699503002441332596121, 5.48925641320362647852716259597, 5.66882303551193056114112541739, 6.17149480136203216358665343108, 6.39804456224879840857164182770, 6.68899086485885057007120434305, 7.07471794438814770083035953668, 7.56380120870672226378041199914, 7.969278943331290246984175117134, 8.509151204991037720525387181518, 8.561690076916054853886704956183
