| L(s) = 1 | − 1.11·2-s + 3-s − 0.758·4-s + 1.17·5-s − 1.11·6-s + 3.07·8-s + 9-s − 1.30·10-s − 0.466·11-s − 0.758·12-s + 1.79·13-s + 1.17·15-s − 1.90·16-s + 0.443·17-s − 1.11·18-s − 2.93·19-s − 0.888·20-s + 0.519·22-s − 23-s + 3.07·24-s − 3.62·25-s − 2.00·26-s + 27-s + 1.64·29-s − 1.30·30-s + 7.66·31-s − 4.01·32-s + ⋯ |
| L(s) = 1 | − 0.787·2-s + 0.577·3-s − 0.379·4-s + 0.523·5-s − 0.454·6-s + 1.08·8-s + 0.333·9-s − 0.412·10-s − 0.140·11-s − 0.218·12-s + 0.498·13-s + 0.302·15-s − 0.477·16-s + 0.107·17-s − 0.262·18-s − 0.672·19-s − 0.198·20-s + 0.110·22-s − 0.208·23-s + 0.627·24-s − 0.725·25-s − 0.393·26-s + 0.192·27-s + 0.305·29-s − 0.238·30-s + 1.37·31-s − 0.710·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.488464106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.488464106\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 1.11T + 2T^{2} \) |
| 5 | \( 1 - 1.17T + 5T^{2} \) |
| 11 | \( 1 + 0.466T + 11T^{2} \) |
| 13 | \( 1 - 1.79T + 13T^{2} \) |
| 17 | \( 1 - 0.443T + 17T^{2} \) |
| 19 | \( 1 + 2.93T + 19T^{2} \) |
| 29 | \( 1 - 1.64T + 29T^{2} \) |
| 31 | \( 1 - 7.66T + 31T^{2} \) |
| 37 | \( 1 - 8.65T + 37T^{2} \) |
| 41 | \( 1 + 1.41T + 41T^{2} \) |
| 43 | \( 1 - 9.81T + 43T^{2} \) |
| 47 | \( 1 - 5.59T + 47T^{2} \) |
| 53 | \( 1 - 3.70T + 53T^{2} \) |
| 59 | \( 1 + 9.83T + 59T^{2} \) |
| 61 | \( 1 + 2.44T + 61T^{2} \) |
| 67 | \( 1 + 9.71T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 3.14T + 73T^{2} \) |
| 79 | \( 1 + 4.92T + 79T^{2} \) |
| 83 | \( 1 - 1.23T + 83T^{2} \) |
| 89 | \( 1 + 2.98T + 89T^{2} \) |
| 97 | \( 1 - 6.32T + 97T^{2} \) |
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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
−8.697656955018765830020483599073, −7.997613687349301475568873639833, −7.49461703359511887111211452718, −6.40876936038806152335800925564, −5.72815033136070670545677498192, −4.57604587821110707651076598914, −4.04778675360473389650087327447, −2.82570435483691589908307410643, −1.88754636002486190681909779785, −0.825516397558115688178841952308,
0.825516397558115688178841952308, 1.88754636002486190681909779785, 2.82570435483691589908307410643, 4.04778675360473389650087327447, 4.57604587821110707651076598914, 5.72815033136070670545677498192, 6.40876936038806152335800925564, 7.49461703359511887111211452718, 7.997613687349301475568873639833, 8.697656955018765830020483599073
