| L(s) = 1 | − 1.41·3-s + 5-s + 2.82·7-s − 0.999·9-s − 0.828·11-s − 3.41·13-s − 1.41·15-s + 4.82·17-s − 19-s − 4.00·21-s − 4·23-s + 25-s + 5.65·27-s + 0.828·29-s + 1.17·33-s + 2.82·35-s − 10.2·37-s + 4.82·39-s − 0.828·41-s + 2.82·43-s − 0.999·45-s + 8.48·47-s + 1.00·49-s − 6.82·51-s − 13.0·53-s − 0.828·55-s + 1.41·57-s + ⋯ |
| L(s) = 1 | − 0.816·3-s + 0.447·5-s + 1.06·7-s − 0.333·9-s − 0.249·11-s − 0.946·13-s − 0.365·15-s + 1.17·17-s − 0.229·19-s − 0.872·21-s − 0.834·23-s + 0.200·25-s + 1.08·27-s + 0.153·29-s + 0.203·33-s + 0.478·35-s − 1.68·37-s + 0.773·39-s − 0.129·41-s + 0.431·43-s − 0.149·45-s + 1.23·47-s + 0.142·49-s − 0.956·51-s − 1.79·53-s − 0.111·55-s + 0.187·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 + 3.41T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 0.828T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 0.828T + 41T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 - 1.65T + 61T^{2} \) |
| 67 | \( 1 + 9.41T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 9.17T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + 3.17T + 89T^{2} \) |
| 97 | \( 1 + 2.24T + 97T^{2} \) |
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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
−7.78513685174352828177882674767, −6.97069244630425161000059216205, −6.15017307823796868359685993137, −5.41413550682614316386569133086, −5.11120029313900633291150555543, −4.28059413026400237982553779563, −3.13566745663642548051411453896, −2.20660523819540497949679362949, −1.29501968330739158341611946759, 0,
1.29501968330739158341611946759, 2.20660523819540497949679362949, 3.13566745663642548051411453896, 4.28059413026400237982553779563, 5.11120029313900633291150555543, 5.41413550682614316386569133086, 6.15017307823796868359685993137, 6.97069244630425161000059216205, 7.78513685174352828177882674767
