| L(s) = 1 | + 2-s + 3-s + 4-s + 3.23·5-s + 6-s + 8-s + 9-s + 3.23·10-s − 11-s + 12-s + 3.23·15-s + 16-s + 0.763·17-s + 18-s − 5.70·19-s + 3.23·20-s − 22-s + 6.47·23-s + 24-s + 5.47·25-s + 27-s − 4.47·29-s + 3.23·30-s + 7.23·31-s + 32-s − 33-s + 0.763·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.44·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 1.02·10-s − 0.301·11-s + 0.288·12-s + 0.835·15-s + 0.250·16-s + 0.185·17-s + 0.235·18-s − 1.30·19-s + 0.723·20-s − 0.213·22-s + 1.34·23-s + 0.204·24-s + 1.09·25-s + 0.192·27-s − 0.830·29-s + 0.590·30-s + 1.29·31-s + 0.176·32-s − 0.174·33-s + 0.131·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.888905848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.888905848\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 7.23T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 - 0.763T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 - 9.70T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 - 2.29T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 - 3.23T + 83T^{2} \) |
| 89 | \( 1 + 2.47T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.741788341757619207695356282185, −7.88558127539567505287852571959, −6.96589949112444028613204410035, −6.30087679671376595225245916209, −5.62760158844231651230699794767, −4.84100887330766401618618991845, −4.00835850179094160703005389683, −2.82355838547928644173038054560, −2.34315154550203202784495590440, −1.30339527837106869667181697221,
1.30339527837106869667181697221, 2.34315154550203202784495590440, 2.82355838547928644173038054560, 4.00835850179094160703005389683, 4.84100887330766401618618991845, 5.62760158844231651230699794767, 6.30087679671376595225245916209, 6.96589949112444028613204410035, 7.88558127539567505287852571959, 8.741788341757619207695356282185
