L(s) = 1 | + 3-s + 5-s + 9-s + 4·11-s − 2·13-s + 15-s − 2·17-s + 4·19-s + 8·23-s + 25-s + 27-s − 6·29-s − 8·31-s + 4·33-s + 2·37-s − 2·39-s − 2·41-s − 12·43-s + 45-s − 8·47-s − 2·51-s − 6·53-s + 4·55-s + 4·57-s − 4·59-s − 2·61-s − 2·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.258·15-s − 0.485·17-s + 0.917·19-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.696·33-s + 0.328·37-s − 0.320·39-s − 0.312·41-s − 1.82·43-s + 0.149·45-s − 1.16·47-s − 0.280·51-s − 0.824·53-s + 0.539·55-s + 0.529·57-s − 0.520·59-s − 0.256·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.73055132525072, −14.44350534478136, −13.89154015472531, −13.21404196181971, −12.99837845867042, −12.37781895872225, −11.67706327958476, −11.24954432620307, −10.80145449410868, −9.939047294934322, −9.445828494950952, −9.277661094272857, −8.664715027438973, −8.009129494398415, −7.355932083564104, −6.812721829178360, −6.533303713345125, −5.544333040276266, −5.114590035228675, −4.496864104752107, −3.624116560092450, −3.298335572743783, −2.501449995274218, −1.678613944989889, −1.266736464060217, 0,
1.266736464060217, 1.678613944989889, 2.501449995274218, 3.298335572743783, 3.624116560092450, 4.496864104752107, 5.114590035228675, 5.544333040276266, 6.533303713345125, 6.812721829178360, 7.355932083564104, 8.009129494398415, 8.664715027438973, 9.277661094272857, 9.445828494950952, 9.939047294934322, 10.80145449410868, 11.24954432620307, 11.67706327958476, 12.37781895872225, 12.99837845867042, 13.21404196181971, 13.89154015472531, 14.44350534478136, 14.73055132525072