| L(s) = 1 | − 0.936·3-s − 3.33·5-s − 7-s − 2.12·9-s − 4.27·11-s − 3.33·13-s + 3.12·15-s + 2·17-s − 0.936·19-s + 0.936·21-s − 3.12·23-s + 6.12·25-s + 4.79·27-s − 1.87·29-s − 6.24·31-s + 3.99·33-s + 3.33·35-s − 1.87·37-s + 3.12·39-s + 12.2·41-s − 4.27·43-s + 7.08·45-s + 49-s − 1.87·51-s + 8.54·53-s + 14.2·55-s + 0.876·57-s + ⋯ |
| L(s) = 1 | − 0.540·3-s − 1.49·5-s − 0.377·7-s − 0.707·9-s − 1.28·11-s − 0.924·13-s + 0.806·15-s + 0.485·17-s − 0.214·19-s + 0.204·21-s − 0.651·23-s + 1.22·25-s + 0.923·27-s − 0.347·29-s − 1.12·31-s + 0.696·33-s + 0.563·35-s − 0.307·37-s + 0.500·39-s + 1.91·41-s − 0.651·43-s + 1.05·45-s + 0.142·49-s − 0.262·51-s + 1.17·53-s + 1.92·55-s + 0.116·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3078850050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3078850050\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 0.936T + 3T^{2} \) |
| 5 | \( 1 + 3.33T + 5T^{2} \) |
| 11 | \( 1 + 4.27T + 11T^{2} \) |
| 13 | \( 1 + 3.33T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 0.936T + 19T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 + 1.87T + 29T^{2} \) |
| 31 | \( 1 + 6.24T + 31T^{2} \) |
| 37 | \( 1 + 1.87T + 37T^{2} \) |
| 41 | \( 1 - 12.2T + 41T^{2} \) |
| 43 | \( 1 + 4.27T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 8.54T + 53T^{2} \) |
| 59 | \( 1 - 7.60T + 59T^{2} \) |
| 61 | \( 1 + 3.33T + 61T^{2} \) |
| 67 | \( 1 + 15.7T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 9.47T + 83T^{2} \) |
| 89 | \( 1 + 0.246T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
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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
−9.218856485195048762553779362359, −8.306063520283419449521538100694, −7.65921843034752959170029649385, −7.14012749393927763155695626067, −5.91991487950114033278448733696, −5.25172061616993512094772609216, −4.33911346829486591503435058394, −3.37586428681008076109563540564, −2.47521103627974456287929866930, −0.36139886898797756619735050082,
0.36139886898797756619735050082, 2.47521103627974456287929866930, 3.37586428681008076109563540564, 4.33911346829486591503435058394, 5.25172061616993512094772609216, 5.91991487950114033278448733696, 7.14012749393927763155695626067, 7.65921843034752959170029649385, 8.306063520283419449521538100694, 9.218856485195048762553779362359
