L(s) = 1 | + (−0.0963 + 0.232i)5-s + (−0.617 + 0.617i)7-s + (−0.505 + 1.21i)11-s + (3.41 − 1.41i)13-s + 2.76·17-s + (−0.189 − 0.458i)19-s + (−4.46 + 4.46i)23-s + (3.49 + 3.49i)25-s + (−0.0101 + 0.00418i)29-s − 4.03i·31-s + (−0.0841 − 0.203i)35-s + (6.30 + 2.61i)37-s + (5.34 + 5.34i)41-s + (10.1 + 4.21i)43-s + 11.5i·47-s + ⋯ |
L(s) = 1 | + (−0.0430 + 0.104i)5-s + (−0.233 + 0.233i)7-s + (−0.152 + 0.367i)11-s + (0.947 − 0.392i)13-s + 0.669·17-s + (−0.0435 − 0.105i)19-s + (−0.931 + 0.931i)23-s + (0.698 + 0.698i)25-s + (−0.00187 + 0.000777i)29-s − 0.724i·31-s + (−0.0142 − 0.0343i)35-s + (1.03 + 0.429i)37-s + (0.834 + 0.834i)41-s + (1.55 + 0.642i)43-s + 1.69i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.570423736\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.570423736\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.0963 - 0.232i)T + (-3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (0.617 - 0.617i)T - 7iT^{2} \) |
| 11 | \( 1 + (0.505 - 1.21i)T + (-7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (-3.41 + 1.41i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 - 2.76T + 17T^{2} \) |
| 19 | \( 1 + (0.189 + 0.458i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (4.46 - 4.46i)T - 23iT^{2} \) |
| 29 | \( 1 + (0.0101 - 0.00418i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 4.03iT - 31T^{2} \) |
| 37 | \( 1 + (-6.30 - 2.61i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (-5.34 - 5.34i)T + 41iT^{2} \) |
| 43 | \( 1 + (-10.1 - 4.21i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 - 11.5iT - 47T^{2} \) |
| 53 | \( 1 + (9.04 + 3.74i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (0.939 + 0.389i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (2.97 + 7.17i)T + (-43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (-7.40 + 3.06i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-1.20 - 1.20i)T + 71iT^{2} \) |
| 73 | \( 1 + (-3.73 + 3.73i)T - 73iT^{2} \) |
| 79 | \( 1 - 7.22T + 79T^{2} \) |
| 83 | \( 1 + (-11.2 + 4.67i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (3.70 - 3.70i)T - 89iT^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−9.605457696443513037905592412136, −9.385859662211237077044158799982, −7.987108201376198381715550511409, −7.70646991426291974748064848090, −6.35723897725603629118028355059, −5.83412416552815764474199466191, −4.71376972075880841330904819509, −3.65095664925916677934259906312, −2.70181443723446105534857556939, −1.23788732589081133982322691218,
0.807937165998463373448341439737, 2.33691200221825026405932244871, 3.57592796596523789513240047224, 4.35737146944742596065616105100, 5.57513452573663909203994101687, 6.30384437152980704271674842832, 7.19111387288078481763649540551, 8.207260238497759478713966754006, 8.768032533898371957027514861644, 9.732694984747034288397948853958