| L(s) = 1 | − 2·2-s + 2·4-s + 3·5-s − 6·10-s − 2·11-s − 6·13-s − 4·16-s + 3·17-s + 6·19-s + 6·20-s + 4·22-s + 8·23-s + 4·25-s + 12·26-s + 2·29-s − 6·31-s + 8·32-s − 6·34-s + 9·37-s − 12·38-s + 9·41-s − 9·43-s − 4·44-s − 16·46-s − 3·47-s − 8·50-s − 12·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1.34·5-s − 1.89·10-s − 0.603·11-s − 1.66·13-s − 16-s + 0.727·17-s + 1.37·19-s + 1.34·20-s + 0.852·22-s + 1.66·23-s + 4/5·25-s + 2.35·26-s + 0.371·29-s − 1.07·31-s + 1.41·32-s − 1.02·34-s + 1.47·37-s − 1.94·38-s + 1.40·41-s − 1.37·43-s − 0.603·44-s − 2.35·46-s − 0.437·47-s − 1.13·50-s − 1.66·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9738724035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9738724035\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−9.479394193370238459103428530459, −9.289581222021464707786603330344, −7.977598033740232460345294759921, −7.44806189292664502395185090799, −6.63733359142638917146855284352, −5.40999706001583504561647394868, −4.90845254944016731114307090228, −2.96534644851181413064575814534, −2.10923233390474501409147906935, −0.913197546741854214881948839395,
0.913197546741854214881948839395, 2.10923233390474501409147906935, 2.96534644851181413064575814534, 4.90845254944016731114307090228, 5.40999706001583504561647394868, 6.63733359142638917146855284352, 7.44806189292664502395185090799, 7.977598033740232460345294759921, 9.289581222021464707786603330344, 9.479394193370238459103428530459
