| L(s) = 1 | + (1.44 − 1.80i)2-s + (−0.745 − 3.26i)4-s + (2.09 − 1.43i)5-s + (−1.09 − 1.89i)7-s + (−2.80 − 1.35i)8-s + (0.438 − 5.85i)10-s + (−0.694 + 3.04i)11-s + (0.257 + 3.43i)13-s + (−5.00 − 0.753i)14-s + (−0.462 + 0.222i)16-s + (1.92 + 1.31i)17-s + (−3.23 − 0.996i)19-s + (−6.23 − 5.78i)20-s + (4.50 + 5.64i)22-s + (−4.82 − 4.47i)23-s + ⋯ |
| L(s) = 1 | + (1.01 − 1.27i)2-s + (−0.372 − 1.63i)4-s + (0.938 − 0.639i)5-s + (−0.413 − 0.715i)7-s + (−0.993 − 0.478i)8-s + (0.138 − 1.85i)10-s + (−0.209 + 0.917i)11-s + (0.0713 + 0.951i)13-s + (−1.33 − 0.201i)14-s + (−0.115 + 0.0557i)16-s + (0.467 + 0.318i)17-s + (−0.741 − 0.228i)19-s + (−1.39 − 1.29i)20-s + (0.959 + 1.20i)22-s + (−1.00 − 0.933i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.10798 - 2.18066i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10798 - 2.18066i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + (-6.03 - 2.55i)T \) |
| good | 2 | \( 1 + (-1.44 + 1.80i)T + (-0.445 - 1.94i)T^{2} \) |
| 5 | \( 1 + (-2.09 + 1.43i)T + (1.82 - 4.65i)T^{2} \) |
| 7 | \( 1 + (1.09 + 1.89i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.694 - 3.04i)T + (-9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (-0.257 - 3.43i)T + (-12.8 + 1.93i)T^{2} \) |
| 17 | \( 1 + (-1.92 - 1.31i)T + (6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (3.23 + 0.996i)T + (15.6 + 10.7i)T^{2} \) |
| 23 | \( 1 + (4.82 + 4.47i)T + (1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-5.34 - 0.806i)T + (27.7 + 8.54i)T^{2} \) |
| 31 | \( 1 + (-0.778 - 1.98i)T + (-22.7 + 21.0i)T^{2} \) |
| 37 | \( 1 + (1.73 - 3.01i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.33 - 1.66i)T + (-9.12 - 39.9i)T^{2} \) |
| 47 | \( 1 + (0.260 + 1.14i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (0.777 - 10.3i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (-5.53 + 2.66i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (-0.913 + 2.32i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-9.87 - 3.04i)T + (55.3 + 37.7i)T^{2} \) |
| 71 | \( 1 + (8.30 - 7.70i)T + (5.30 - 70.8i)T^{2} \) |
| 73 | \( 1 + (-0.624 - 8.33i)T + (-72.1 + 10.8i)T^{2} \) |
| 79 | \( 1 + (4.90 + 8.50i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.83 + 0.427i)T + (79.3 - 24.4i)T^{2} \) |
| 89 | \( 1 + (14.8 - 2.23i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (-0.939 + 4.11i)T + (-87.3 - 42.0i)T^{2} \) |
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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
−11.06264589808943899004302208963, −10.09635104597292242468191844243, −9.799769277372258913168777950021, −8.509007676695194931492308897609, −6.92058565293717674121442506731, −5.86161224896771440828966206054, −4.69696193450130653126572599670, −4.06588498297651542229841220709, −2.52188884025354957203579629952, −1.45048726503893462755219099394,
2.62885444145153024080092698212, 3.72530625347663056994933530088, 5.34670757777242922147080321638, 5.88494493298193865606580962534, 6.49866996883573397896142437592, 7.71142138486693966777817482967, 8.547089275898775310741246912342, 9.816234738827523537895749075523, 10.65016352698630265232986709033, 12.00792028460764213260636713982
