| L(s) = 1 | + (−7.75 − 4.47i)5-s + (−5.17 − 4.71i)7-s + (−1.91 + 1.10i)11-s + 15.7·13-s + (−15.1 + 8.72i)17-s + (−1.79 + 3.10i)19-s + (6.65 + 3.84i)23-s + (27.5 + 47.7i)25-s + 24.6i·29-s + (22.9 + 39.7i)31-s + (19.0 + 59.7i)35-s + (−2.46 + 4.26i)37-s − 28.9i·41-s − 35.9·43-s + (−41.2 − 23.7i)47-s + ⋯ |
| L(s) = 1 | + (−1.55 − 0.895i)5-s + (−0.739 − 0.673i)7-s + (−0.174 + 0.100i)11-s + 1.21·13-s + (−0.889 + 0.513i)17-s + (−0.0943 + 0.163i)19-s + (0.289 + 0.167i)23-s + (1.10 + 1.91i)25-s + 0.848i·29-s + (0.740 + 1.28i)31-s + (0.543 + 1.70i)35-s + (−0.0665 + 0.115i)37-s − 0.705i·41-s − 0.836·43-s + (−0.877 − 0.506i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0601 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0601 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4820745781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4820745781\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (5.17 + 4.71i)T \) |
| good | 5 | \( 1 + (7.75 + 4.47i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (1.91 - 1.10i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 15.7T + 169T^{2} \) |
| 17 | \( 1 + (15.1 - 8.72i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (1.79 - 3.10i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-6.65 - 3.84i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 24.6iT - 841T^{2} \) |
| 31 | \( 1 + (-22.9 - 39.7i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (2.46 - 4.26i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 28.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 35.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + (41.2 + 23.7i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-50.1 + 28.9i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (100. - 58.0i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-53.9 + 93.4i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-65.9 - 114. i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 115. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (23.1 + 40.1i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (10.0 - 17.4i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 5.93iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (67.5 + 38.9i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 87.3T + 9.40e3T^{2} \) |
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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
−11.01477117100383323239292487409, −10.15862985854577370967161247578, −8.758592319201683469760429044004, −8.461465440241859480386274357937, −7.31106549409327113104369370522, −6.51702050163950141979794173282, −5.05310595724380094242464556080, −4.04166425988673150279703946484, −3.38649774804915748130079699820, −1.13362049357232106480795327316,
0.22735830136323886609991480318, 2.64643607578802991208614708542, 3.52812928233158072211719623723, 4.51874140993263418532948175458, 6.12936184887660237469624093126, 6.75766474348474716567304879254, 7.84934762079110561200516628421, 8.555456991159952918116948346657, 9.591752852471688082983166319545, 10.75751442009441603412636873930
