| L(s) = 1 | − 2.71·3-s − 3.85·5-s + 4.39·9-s − 0.245·11-s + 13-s + 10.4·15-s − 7.16·17-s + 5.01·19-s − 7.80·23-s + 9.87·25-s − 3.80·27-s + 7.54·29-s + 5.89·31-s + 0.668·33-s + 9.24·37-s − 2.71·39-s + 6.63·41-s − 0.559·43-s − 16.9·45-s − 2.69·47-s + 19.4·51-s − 6.48·53-s + 0.947·55-s − 13.6·57-s + 8.50·59-s − 1.89·61-s − 3.85·65-s + ⋯ |
| L(s) = 1 | − 1.57·3-s − 1.72·5-s + 1.46·9-s − 0.0740·11-s + 0.277·13-s + 2.70·15-s − 1.73·17-s + 1.15·19-s − 1.62·23-s + 1.97·25-s − 0.731·27-s + 1.40·29-s + 1.05·31-s + 0.116·33-s + 1.51·37-s − 0.435·39-s + 1.03·41-s − 0.0853·43-s − 2.52·45-s − 0.393·47-s + 2.72·51-s − 0.890·53-s + 0.127·55-s − 1.80·57-s + 1.10·59-s − 0.242·61-s − 0.478·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2.71T + 3T^{2} \) |
| 5 | \( 1 + 3.85T + 5T^{2} \) |
| 11 | \( 1 + 0.245T + 11T^{2} \) |
| 17 | \( 1 + 7.16T + 17T^{2} \) |
| 19 | \( 1 - 5.01T + 19T^{2} \) |
| 23 | \( 1 + 7.80T + 23T^{2} \) |
| 29 | \( 1 - 7.54T + 29T^{2} \) |
| 31 | \( 1 - 5.89T + 31T^{2} \) |
| 37 | \( 1 - 9.24T + 37T^{2} \) |
| 41 | \( 1 - 6.63T + 41T^{2} \) |
| 43 | \( 1 + 0.559T + 43T^{2} \) |
| 47 | \( 1 + 2.69T + 47T^{2} \) |
| 53 | \( 1 + 6.48T + 53T^{2} \) |
| 59 | \( 1 - 8.50T + 59T^{2} \) |
| 61 | \( 1 + 1.89T + 61T^{2} \) |
| 67 | \( 1 + 0.851T + 67T^{2} \) |
| 71 | \( 1 + 1.08T + 71T^{2} \) |
| 73 | \( 1 + 1.71T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 9.31T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 - 0.00695T + 97T^{2} \) |
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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
−8.200506704201386222534068382604, −7.81870986971578044419146512616, −6.73974125403333887990867454334, −6.38598522781211034474968259008, −5.32132204949714686236038376502, −4.39773478517033842729343596089, −4.13561388233245189716375177772, −2.76609825619625087609745839128, −0.973210217265114528184785970518, 0,
0.973210217265114528184785970518, 2.76609825619625087609745839128, 4.13561388233245189716375177772, 4.39773478517033842729343596089, 5.32132204949714686236038376502, 6.38598522781211034474968259008, 6.73974125403333887990867454334, 7.81870986971578044419146512616, 8.200506704201386222534068382604
