| L(s) = 1 | + 3.24·2-s − 3·3-s + 2.53·4-s − 17.4·5-s − 9.73·6-s − 26.6·7-s − 17.7·8-s + 9·9-s − 56.6·10-s + 11·11-s − 7.60·12-s + 89.8·13-s − 86.3·14-s + 52.3·15-s − 77.8·16-s + 47.6·17-s + 29.2·18-s + 60.0·19-s − 44.2·20-s + 79.8·21-s + 35.7·22-s + 13.3·23-s + 53.2·24-s + 179.·25-s + 291.·26-s − 27·27-s − 67.4·28-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 0.577·3-s + 0.316·4-s − 1.56·5-s − 0.662·6-s − 1.43·7-s − 0.783·8-s + 0.333·9-s − 1.79·10-s + 0.301·11-s − 0.182·12-s + 1.91·13-s − 1.64·14-s + 0.901·15-s − 1.21·16-s + 0.679·17-s + 0.382·18-s + 0.725·19-s − 0.494·20-s + 0.829·21-s + 0.345·22-s + 0.121·23-s + 0.452·24-s + 1.43·25-s + 2.20·26-s − 0.192·27-s − 0.455·28-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 3.24T + 8T^{2} \) |
| 5 | \( 1 + 17.4T + 125T^{2} \) |
| 7 | \( 1 + 26.6T + 343T^{2} \) |
| 13 | \( 1 - 89.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 47.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 60.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 13.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 48.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 361.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 57.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 29.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 223.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 165.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 84.1T + 2.05e5T^{2} \) |
| 67 | \( 1 + 365.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 682.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.05e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 149.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 397.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 165.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 405.T + 9.12e5T^{2} \) |
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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
−8.411553109664289244011175909995, −7.36225007394722763500447271354, −6.52963886023440913084900139887, −6.03287159280739512723978226470, −5.11451899111767227802166857013, −4.08981750014009309891643829880, −3.55037036449492766387868901737, −3.14169809554196933483382340011, −0.986258231253848758157296295475, 0,
0.986258231253848758157296295475, 3.14169809554196933483382340011, 3.55037036449492766387868901737, 4.08981750014009309891643829880, 5.11451899111767227802166857013, 6.03287159280739512723978226470, 6.52963886023440913084900139887, 7.36225007394722763500447271354, 8.411553109664289244011175909995
