| L(s) = 1 | − 5-s − 2·11-s − 5·13-s − 4·17-s + 3·19-s − 4·23-s + 25-s + 4·29-s + 31-s + 37-s − 6·41-s − 43-s + 10·47-s + 2·55-s + 6·61-s + 5·65-s − 15·67-s − 6·71-s − 3·73-s − 9·79-s − 6·83-s + 4·85-s − 3·95-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.603·11-s − 1.38·13-s − 0.970·17-s + 0.688·19-s − 0.834·23-s + 1/5·25-s + 0.742·29-s + 0.179·31-s + 0.164·37-s − 0.937·41-s − 0.152·43-s + 1.45·47-s + 0.269·55-s + 0.768·61-s + 0.620·65-s − 1.83·67-s − 0.712·71-s − 0.351·73-s − 1.01·79-s − 0.658·83-s + 0.433·85-s − 0.307·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.69500808606238, −13.14412113212367, −12.62723627864054, −12.18742423387962, −11.68086393112686, −11.47708738215253, −10.62352010474000, −10.29203122313386, −9.891978551611036, −9.316543951377754, −8.679146789573584, −8.373898527006340, −7.646056280057469, −7.309152640655875, −6.947316767462538, −6.162790510547003, −5.704903601565911, −5.028003499749839, −4.581828725447588, −4.195286555267838, −3.363391469085040, −2.803746242900701, −2.320884189647981, −1.644983590577131, −0.6497823583133864, 0,
0.6497823583133864, 1.644983590577131, 2.320884189647981, 2.803746242900701, 3.363391469085040, 4.195286555267838, 4.581828725447588, 5.028003499749839, 5.704903601565911, 6.162790510547003, 6.947316767462538, 7.309152640655875, 7.646056280057469, 8.373898527006340, 8.679146789573584, 9.316543951377754, 9.891978551611036, 10.29203122313386, 10.62352010474000, 11.47708738215253, 11.68086393112686, 12.18742423387962, 12.62723627864054, 13.14412113212367, 13.69500808606238
