| L(s) = 1 | + 2.11·2-s + 3-s + 2.47·4-s + 1.64·5-s + 2.11·6-s + 1.00·8-s + 9-s + 3.47·10-s + 11-s + 2.47·12-s + 4.58·13-s + 1.64·15-s − 2.83·16-s + 0.715·17-s + 2.11·18-s + 1.64·19-s + 4.06·20-s + 2.11·22-s + 1.00·24-s − 2.30·25-s + 9.70·26-s + 27-s − 5.53·29-s + 3.47·30-s − 5.17·31-s − 7.98·32-s + 33-s + ⋯ |
| L(s) = 1 | + 1.49·2-s + 0.577·3-s + 1.23·4-s + 0.734·5-s + 0.863·6-s + 0.353·8-s + 0.333·9-s + 1.09·10-s + 0.301·11-s + 0.713·12-s + 1.27·13-s + 0.423·15-s − 0.707·16-s + 0.173·17-s + 0.498·18-s + 0.376·19-s + 0.907·20-s + 0.450·22-s + 0.204·24-s − 0.460·25-s + 1.90·26-s + 0.192·27-s − 1.02·29-s + 0.634·30-s − 0.929·31-s − 1.41·32-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.425380747\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.425380747\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.11T + 2T^{2} \) |
| 5 | \( 1 - 1.64T + 5T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 17 | \( 1 - 0.715T + 17T^{2} \) |
| 19 | \( 1 - 1.64T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 5.53T + 29T^{2} \) |
| 31 | \( 1 + 5.17T + 31T^{2} \) |
| 37 | \( 1 + 6.58T + 37T^{2} \) |
| 41 | \( 1 - 5.17T + 41T^{2} \) |
| 43 | \( 1 - 5.17T + 43T^{2} \) |
| 47 | \( 1 - 7.87T + 47T^{2} \) |
| 53 | \( 1 - 2.71T + 53T^{2} \) |
| 59 | \( 1 + 3.15T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 4.21T + 67T^{2} \) |
| 71 | \( 1 + 1.85T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 9.17T + 83T^{2} \) |
| 89 | \( 1 - 4.71T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−9.249231668232122523465801106157, −8.779268319986869671551412957382, −7.57153378145971643248515349007, −6.74618635002000132020893689462, −5.79759602624576880994992442727, −5.46240768225880472051407318515, −4.12301393352255220788027491827, −3.65665845413847059418412007617, −2.60946159557131001245877039894, −1.58778216428596393451246135794,
1.58778216428596393451246135794, 2.60946159557131001245877039894, 3.65665845413847059418412007617, 4.12301393352255220788027491827, 5.46240768225880472051407318515, 5.79759602624576880994992442727, 6.74618635002000132020893689462, 7.57153378145971643248515349007, 8.779268319986869671551412957382, 9.249231668232122523465801106157
