| L(s) = 1 | − 5.27·2-s − 3·3-s + 19.8·4-s − 1.30·5-s + 15.8·6-s + 18.3·7-s − 62.4·8-s + 9·9-s + 6.85·10-s − 11·11-s − 59.5·12-s − 9.87·13-s − 97.0·14-s + 3.90·15-s + 170.·16-s − 113.·17-s − 47.4·18-s + 80.8·19-s − 25.7·20-s − 55.1·21-s + 58.0·22-s − 68.3·23-s + 187.·24-s − 123.·25-s + 52.0·26-s − 27·27-s + 364.·28-s + ⋯ |
| L(s) = 1 | − 1.86·2-s − 0.577·3-s + 2.47·4-s − 0.116·5-s + 1.07·6-s + 0.993·7-s − 2.76·8-s + 0.333·9-s + 0.216·10-s − 0.301·11-s − 1.43·12-s − 0.210·13-s − 1.85·14-s + 0.0671·15-s + 2.66·16-s − 1.61·17-s − 0.621·18-s + 0.976·19-s − 0.288·20-s − 0.573·21-s + 0.562·22-s − 0.620·23-s + 1.59·24-s − 0.986·25-s + 0.392·26-s − 0.192·27-s + 2.46·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3111175272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3111175272\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 + 5.27T + 8T^{2} \) |
| 5 | \( 1 + 1.30T + 125T^{2} \) |
| 7 | \( 1 - 18.3T + 343T^{2} \) |
| 13 | \( 1 + 9.87T + 2.19e3T^{2} \) |
| 17 | \( 1 + 113.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 80.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 68.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 271.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 240.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 70.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 322.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 153.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 243.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 531.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 316.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 830.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.13e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 650.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 534.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 745.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 767.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.17e3T + 9.12e5T^{2} \) |
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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
−8.924233588226787132781937698465, −7.947283392928416036067060910378, −7.53863588662329275305669896967, −6.79894472819353250257450370950, −5.84626708245523362711210373952, −4.99841984691895245871831057689, −3.69059539281567962785414839773, −2.14520807633222234587889191933, −1.68222339799902422598772683956, −0.34479071352947815691939542505,
0.34479071352947815691939542505, 1.68222339799902422598772683956, 2.14520807633222234587889191933, 3.69059539281567962785414839773, 4.99841984691895245871831057689, 5.84626708245523362711210373952, 6.79894472819353250257450370950, 7.53863588662329275305669896967, 7.947283392928416036067060910378, 8.924233588226787132781937698465
