L(s) = 1 | − 0.275·2-s + 3.60·3-s − 7.92·4-s − 3.79·5-s − 0.995·6-s + 4.39·8-s − 13.9·9-s + 1.04·10-s − 55.4·11-s − 28.6·12-s + 48.3·13-s − 13.7·15-s + 62.1·16-s + 57.3·17-s + 3.85·18-s − 112.·19-s + 30.1·20-s + 15.3·22-s + 120.·23-s + 15.8·24-s − 110.·25-s − 13.3·26-s − 147.·27-s + 171.·29-s + 3.78·30-s − 75.8·31-s − 52.3·32-s + ⋯ |
L(s) = 1 | − 0.0975·2-s + 0.694·3-s − 0.990·4-s − 0.339·5-s − 0.0677·6-s + 0.194·8-s − 0.517·9-s + 0.0331·10-s − 1.52·11-s − 0.688·12-s + 1.03·13-s − 0.236·15-s + 0.971·16-s + 0.818·17-s + 0.0504·18-s − 1.35·19-s + 0.336·20-s + 0.148·22-s + 1.09·23-s + 0.134·24-s − 0.884·25-s − 0.100·26-s − 1.05·27-s + 1.09·29-s + 0.0230·30-s − 0.439·31-s − 0.288·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.9387077424\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9387077424\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
good | 2 | \( 1 + 0.275T + 8T^{2} \) |
| 3 | \( 1 - 3.60T + 27T^{2} \) |
| 5 | \( 1 + 3.79T + 125T^{2} \) |
| 11 | \( 1 + 55.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 48.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 57.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 112.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 120.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 171.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 75.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 397.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 86.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 47.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 416.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 315.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 689.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 245.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 367.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 91.8T + 3.57e5T^{2} \) |
| 73 | \( 1 + 55.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 436.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.29e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 77.1T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.26e3T + 9.12e5T^{2} \) |
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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
−8.340011691557310865687016637123, −8.296928364537793738196632522139, −7.39588678296730869860986909577, −6.17094887576870837478474769614, −5.35229100002721212415724510735, −4.64075594015586938795353202873, −3.55028037838506806825551047655, −3.07101097431069297507440416900, −1.80562448326619056658479891093, −0.41375195975288157120089561023,
0.41375195975288157120089561023, 1.80562448326619056658479891093, 3.07101097431069297507440416900, 3.55028037838506806825551047655, 4.64075594015586938795353202873, 5.35229100002721212415724510735, 6.17094887576870837478474769614, 7.39588678296730869860986909577, 8.296928364537793738196632522139, 8.340011691557310865687016637123