| L(s) = 1 | + 5.35·2-s − 3·3-s + 20.7·4-s − 16.0·6-s + 4.43·7-s + 68.1·8-s + 9·9-s − 3.43·11-s − 62.1·12-s − 78.7·13-s + 23.7·14-s + 199.·16-s − 53.1·17-s + 48.2·18-s + 20.4·19-s − 13.3·21-s − 18.4·22-s − 118.·23-s − 204.·24-s − 421.·26-s − 27·27-s + 91.8·28-s + 168.·29-s − 61.0·31-s + 523.·32-s + 10.3·33-s − 284.·34-s + ⋯ |
| L(s) = 1 | + 1.89·2-s − 0.577·3-s + 2.58·4-s − 1.09·6-s + 0.239·7-s + 3.01·8-s + 0.333·9-s − 0.0941·11-s − 1.49·12-s − 1.67·13-s + 0.453·14-s + 3.11·16-s − 0.758·17-s + 0.631·18-s + 0.246·19-s − 0.138·21-s − 0.178·22-s − 1.07·23-s − 1.73·24-s − 3.18·26-s − 0.192·27-s + 0.620·28-s + 1.07·29-s − 0.353·31-s + 2.89·32-s + 0.0543·33-s − 1.43·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.513539501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.513539501\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 5.35T + 8T^{2} \) |
| 7 | \( 1 - 4.43T + 343T^{2} \) |
| 11 | \( 1 + 3.43T + 1.33e3T^{2} \) |
| 13 | \( 1 + 78.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 53.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 20.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 118.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 168.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 61.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 246.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 422.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 362.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 170.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 546.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 216.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 130.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 614.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 324.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 88.8T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 758.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 195.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 521T + 9.12e5T^{2} \) |
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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
−14.03788744922045041459937948761, −12.89573094209689466971672645536, −12.09777727215669085354721759107, −11.28826108147487817190758028332, −10.05857301981669474608956321730, −7.59758035965249877213244615909, −6.47057889943727675642661651267, −5.23797604469871097058329697238, −4.29056976609640423991413652363, −2.42335879264394898791453723953,
2.42335879264394898791453723953, 4.29056976609640423991413652363, 5.23797604469871097058329697238, 6.47057889943727675642661651267, 7.59758035965249877213244615909, 10.05857301981669474608956321730, 11.28826108147487817190758028332, 12.09777727215669085354721759107, 12.89573094209689466971672645536, 14.03788744922045041459937948761
