| L(s) = 1 | − 2-s − 2.92·3-s + 4-s + 0.618·5-s + 2.92·6-s − 2.50·7-s − 8-s + 5.55·9-s − 0.618·10-s + 1.73·11-s − 2.92·12-s − 4.92·13-s + 2.50·14-s − 1.81·15-s + 16-s − 6.02·17-s − 5.55·18-s − 2.48·19-s + 0.618·20-s + 7.32·21-s − 1.73·22-s + 23-s + 2.92·24-s − 4.61·25-s + 4.92·26-s − 7.47·27-s − 2.50·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.68·3-s + 0.5·4-s + 0.276·5-s + 1.19·6-s − 0.945·7-s − 0.353·8-s + 1.85·9-s − 0.195·10-s + 0.522·11-s − 0.844·12-s − 1.36·13-s + 0.668·14-s − 0.467·15-s + 0.250·16-s − 1.46·17-s − 1.30·18-s − 0.569·19-s + 0.138·20-s + 1.59·21-s − 0.369·22-s + 0.208·23-s + 0.597·24-s − 0.923·25-s + 0.966·26-s − 1.43·27-s − 0.472·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05052458029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05052458029\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 + 2.92T + 3T^{2} \) |
| 5 | \( 1 - 0.618T + 5T^{2} \) |
| 7 | \( 1 + 2.50T + 7T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 + 4.92T + 13T^{2} \) |
| 17 | \( 1 + 6.02T + 17T^{2} \) |
| 19 | \( 1 + 2.48T + 19T^{2} \) |
| 29 | \( 1 + 7.80T + 29T^{2} \) |
| 31 | \( 1 + 3.86T + 31T^{2} \) |
| 37 | \( 1 - 4.95T + 37T^{2} \) |
| 41 | \( 1 - 7.29T + 41T^{2} \) |
| 43 | \( 1 + 5.98T + 43T^{2} \) |
| 47 | \( 1 + 9.67T + 47T^{2} \) |
| 53 | \( 1 + 0.792T + 53T^{2} \) |
| 59 | \( 1 - 2.84T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 8.47T + 71T^{2} \) |
| 73 | \( 1 - 5.16T + 73T^{2} \) |
| 79 | \( 1 - 9.76T + 79T^{2} \) |
| 83 | \( 1 - 8.24T + 83T^{2} \) |
| 89 | \( 1 + 5.87T + 89T^{2} \) |
| 97 | \( 1 - 0.455T + 97T^{2} \) |
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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
−7.85996722991529176104241787817, −7.19752906531913287841181812754, −6.44799842576432985208054291134, −6.27997819348875516078314338260, −5.37002004792925168029951746694, −4.63482575415697115638003274579, −3.78427351917964393374828932430, −2.47525221565480393851482527314, −1.60487567927255966682098917302, −0.14020331286454308599452622408,
0.14020331286454308599452622408, 1.60487567927255966682098917302, 2.47525221565480393851482527314, 3.78427351917964393374828932430, 4.63482575415697115638003274579, 5.37002004792925168029951746694, 6.27997819348875516078314338260, 6.44799842576432985208054291134, 7.19752906531913287841181812754, 7.85996722991529176104241787817
