| L(s) = 1 | − 0.0730·3-s − 4.10·5-s − 1.52·7-s − 2.99·9-s + 0.141·11-s + 5.15·13-s + 0.300·15-s − 0.728·17-s + 4.77·19-s + 0.111·21-s − 3.29·23-s + 11.8·25-s + 0.438·27-s + 5.05·29-s + 6.69·31-s − 0.0103·33-s + 6.25·35-s + 0.889·37-s − 0.376·39-s − 3.88·41-s + 10.3·43-s + 12.3·45-s − 10.1·47-s − 4.68·49-s + 0.0532·51-s − 5.37·53-s − 0.580·55-s + ⋯ |
| L(s) = 1 | − 0.0421·3-s − 1.83·5-s − 0.575·7-s − 0.998·9-s + 0.0426·11-s + 1.43·13-s + 0.0775·15-s − 0.176·17-s + 1.09·19-s + 0.0242·21-s − 0.687·23-s + 2.37·25-s + 0.0842·27-s + 0.938·29-s + 1.20·31-s − 0.00179·33-s + 1.05·35-s + 0.146·37-s − 0.0603·39-s − 0.607·41-s + 1.57·43-s + 1.83·45-s − 1.47·47-s − 0.669·49-s + 0.00745·51-s − 0.738·53-s − 0.0783·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 0.0730T + 3T^{2} \) |
| 5 | \( 1 + 4.10T + 5T^{2} \) |
| 7 | \( 1 + 1.52T + 7T^{2} \) |
| 11 | \( 1 - 0.141T + 11T^{2} \) |
| 13 | \( 1 - 5.15T + 13T^{2} \) |
| 17 | \( 1 + 0.728T + 17T^{2} \) |
| 19 | \( 1 - 4.77T + 19T^{2} \) |
| 23 | \( 1 + 3.29T + 23T^{2} \) |
| 29 | \( 1 - 5.05T + 29T^{2} \) |
| 31 | \( 1 - 6.69T + 31T^{2} \) |
| 37 | \( 1 - 0.889T + 37T^{2} \) |
| 41 | \( 1 + 3.88T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 5.37T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 2.06T + 61T^{2} \) |
| 67 | \( 1 - 0.840T + 67T^{2} \) |
| 71 | \( 1 + 3.16T + 71T^{2} \) |
| 73 | \( 1 - 1.65T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 + 4.48T + 83T^{2} \) |
| 89 | \( 1 - 9.40T + 89T^{2} \) |
| 97 | \( 1 - 3.59T + 97T^{2} \) |
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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
−8.169583163355947729796301335376, −7.51758681731177609813014537240, −6.54483647430901403606670543462, −6.04739777735334066988313290631, −4.92762343723578421140116812949, −4.14558885419495160828881243036, −3.33848977736958592998655546308, −2.93096670175720945138633799714, −1.09163364418370873198373519167, 0,
1.09163364418370873198373519167, 2.93096670175720945138633799714, 3.33848977736958592998655546308, 4.14558885419495160828881243036, 4.92762343723578421140116812949, 6.04739777735334066988313290631, 6.54483647430901403606670543462, 7.51758681731177609813014537240, 8.169583163355947729796301335376
