| L(s) = 1 | + 1.52·2-s − 2.27·3-s − 5.67·4-s − 3.47·6-s − 9.53·7-s − 20.8·8-s − 21.8·9-s + 58.3·11-s + 12.9·12-s + 68.7·13-s − 14.5·14-s + 13.6·16-s − 87.2·17-s − 33.2·18-s − 122.·19-s + 21.7·21-s + 88.8·22-s − 181.·23-s + 47.5·24-s + 104.·26-s + 111.·27-s + 54.1·28-s + 39.0·29-s − 159.·31-s + 187.·32-s − 133.·33-s − 132.·34-s + ⋯ |
| L(s) = 1 | + 0.538·2-s − 0.438·3-s − 0.709·4-s − 0.236·6-s − 0.514·7-s − 0.921·8-s − 0.807·9-s + 1.59·11-s + 0.311·12-s + 1.46·13-s − 0.277·14-s + 0.213·16-s − 1.24·17-s − 0.434·18-s − 1.47·19-s + 0.225·21-s + 0.861·22-s − 1.64·23-s + 0.404·24-s + 0.790·26-s + 0.793·27-s + 0.365·28-s + 0.250·29-s − 0.924·31-s + 1.03·32-s − 0.701·33-s − 0.670·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2075 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9681457403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9681457403\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 83 | \( 1 + 83T \) |
| good | 2 | \( 1 - 1.52T + 8T^{2} \) |
| 3 | \( 1 + 2.27T + 27T^{2} \) |
| 7 | \( 1 + 9.53T + 343T^{2} \) |
| 11 | \( 1 - 58.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 68.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 87.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 181.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 39.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 159.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 174.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 232.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 268.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 373.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 532.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 492.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 324.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 583.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.04e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 510.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.19e3T + 4.93e5T^{2} \) |
| 89 | \( 1 + 170.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 464.T + 9.12e5T^{2} \) |
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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
−8.704436453921619449317074888410, −8.381647769239456050155702486969, −6.78152311450078926801975055198, −6.03909203445334219327077330094, −5.96340754682630890569270256903, −4.45644220510339038874352103562, −4.04735581890624827895758833551, −3.20359007425552519783863735212, −1.80358843520486377309104932889, −0.41937864186674522534631694724,
0.41937864186674522534631694724, 1.80358843520486377309104932889, 3.20359007425552519783863735212, 4.04735581890624827895758833551, 4.45644220510339038874352103562, 5.96340754682630890569270256903, 6.03909203445334219327077330094, 6.78152311450078926801975055198, 8.381647769239456050155702486969, 8.704436453921619449317074888410
