| L(s) = 1 | + 3·3-s + 0.612·5-s − 22.7·7-s + 9·9-s − 60.2·11-s + 52.9·13-s + 1.83·15-s + 47.1·17-s − 29.1·19-s − 68.2·21-s + 109.·23-s − 124.·25-s + 27·27-s − 10.4·29-s + 220.·31-s − 180.·33-s − 13.9·35-s + 408.·37-s + 158.·39-s + 360.·41-s + 236.·43-s + 5.51·45-s − 129.·47-s + 174.·49-s + 141.·51-s + 117.·53-s − 36.9·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.0547·5-s − 1.22·7-s + 0.333·9-s − 1.65·11-s + 1.12·13-s + 0.0316·15-s + 0.672·17-s − 0.352·19-s − 0.709·21-s + 0.992·23-s − 0.996·25-s + 0.192·27-s − 0.0667·29-s + 1.27·31-s − 0.953·33-s − 0.0672·35-s + 1.81·37-s + 0.651·39-s + 1.37·41-s + 0.838·43-s + 0.0182·45-s − 0.400·47-s + 0.508·49-s + 0.388·51-s + 0.305·53-s − 0.0905·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.014305061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.014305061\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| good | 5 | \( 1 - 0.612T + 125T^{2} \) |
| 7 | \( 1 + 22.7T + 343T^{2} \) |
| 11 | \( 1 + 60.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 52.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 47.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 29.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 109.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 10.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 220.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 408.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 360.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 236.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 129.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 117.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 262.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 273.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 89.4T + 3.00e5T^{2} \) |
| 71 | \( 1 + 350.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 532.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 166.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 361.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 40.3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 614.T + 9.12e5T^{2} \) |
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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
−9.915656338882297726604983417962, −9.138767165061614496104429642352, −8.159205428791542731130994177882, −7.51961821174401795845188287536, −6.34776366173089140952924395936, −5.64033945901293393284958228663, −4.30292988943011918607781189934, −3.20465985923510888784387052689, −2.50522096875328354790343445610, −0.76773797028134064576551255095,
0.76773797028134064576551255095, 2.50522096875328354790343445610, 3.20465985923510888784387052689, 4.30292988943011918607781189934, 5.64033945901293393284958228663, 6.34776366173089140952924395936, 7.51961821174401795845188287536, 8.159205428791542731130994177882, 9.138767165061614496104429642352, 9.915656338882297726604983417962
