| L(s) = 1 | − 3·3-s + 9.01·5-s + 7.78·7-s + 9·9-s − 15.2·11-s − 13·13-s − 27.0·15-s + 116.·17-s − 79.0·19-s − 23.3·21-s − 30.2·23-s − 43.7·25-s − 27·27-s + 217.·29-s − 250.·31-s + 45.8·33-s + 70.1·35-s − 360.·37-s + 39·39-s + 185.·41-s + 407.·43-s + 81.1·45-s − 34.8·47-s − 282.·49-s − 350.·51-s − 196.·53-s − 137.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.806·5-s + 0.420·7-s + 0.333·9-s − 0.418·11-s − 0.277·13-s − 0.465·15-s + 1.66·17-s − 0.953·19-s − 0.242·21-s − 0.273·23-s − 0.349·25-s − 0.192·27-s + 1.39·29-s − 1.45·31-s + 0.241·33-s + 0.339·35-s − 1.60·37-s + 0.160·39-s + 0.706·41-s + 1.44·43-s + 0.268·45-s − 0.108·47-s − 0.823·49-s − 0.963·51-s − 0.509·53-s − 0.337·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 - 9.01T + 125T^{2} \) |
| 7 | \( 1 - 7.78T + 343T^{2} \) |
| 11 | \( 1 + 15.2T + 1.33e3T^{2} \) |
| 17 | \( 1 - 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 79.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 30.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 217.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 250.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 360.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 185.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 407.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 34.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 196.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 229.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 385.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.06e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 311.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 288.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 160.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 12.2T + 9.12e5T^{2} \) |
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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
−8.050257406010622338138062398687, −7.48989982145016633755559514670, −6.48423478719681905386795249197, −5.76856843707421056041550429765, −5.22240054729020231173767246436, −4.35260271623941808680235785580, −3.23033546624706861457632054105, −2.11311075579362643650808931241, −1.28052578447151324338181488701, 0,
1.28052578447151324338181488701, 2.11311075579362643650808931241, 3.23033546624706861457632054105, 4.35260271623941808680235785580, 5.22240054729020231173767246436, 5.76856843707421056041550429765, 6.48423478719681905386795249197, 7.48989982145016633755559514670, 8.050257406010622338138062398687
