L(s) = 1 | + (2.98 + 2.98i)5-s + 7-s + (−2.32 + 2.32i)11-s + (1.27 + 1.27i)13-s − 3.56i·17-s + (0.796 − 0.796i)19-s − 1.75i·23-s + 12.8i·25-s + (1.87 − 1.87i)29-s + 7.15i·31-s + (2.98 + 2.98i)35-s + (4.64 − 4.64i)37-s − 8.98·41-s + (6.04 + 6.04i)43-s + 6.99·47-s + ⋯ |
L(s) = 1 | + (1.33 + 1.33i)5-s + 0.377·7-s + (−0.701 + 0.701i)11-s + (0.354 + 0.354i)13-s − 0.863i·17-s + (0.182 − 0.182i)19-s − 0.366i·23-s + 2.57i·25-s + (0.348 − 0.348i)29-s + 1.28i·31-s + (0.505 + 0.505i)35-s + (0.762 − 0.762i)37-s − 1.40·41-s + (0.921 + 0.921i)43-s + 1.01·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0647 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0647 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.470071668\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.470071668\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + (-2.98 - 2.98i)T + 5iT^{2} \) |
| 11 | \( 1 + (2.32 - 2.32i)T - 11iT^{2} \) |
| 13 | \( 1 + (-1.27 - 1.27i)T + 13iT^{2} \) |
| 17 | \( 1 + 3.56iT - 17T^{2} \) |
| 19 | \( 1 + (-0.796 + 0.796i)T - 19iT^{2} \) |
| 23 | \( 1 + 1.75iT - 23T^{2} \) |
| 29 | \( 1 + (-1.87 + 1.87i)T - 29iT^{2} \) |
| 31 | \( 1 - 7.15iT - 31T^{2} \) |
| 37 | \( 1 + (-4.64 + 4.64i)T - 37iT^{2} \) |
| 41 | \( 1 + 8.98T + 41T^{2} \) |
| 43 | \( 1 + (-6.04 - 6.04i)T + 43iT^{2} \) |
| 47 | \( 1 - 6.99T + 47T^{2} \) |
| 53 | \( 1 + (0.536 + 0.536i)T + 53iT^{2} \) |
| 59 | \( 1 + (0.119 - 0.119i)T - 59iT^{2} \) |
| 61 | \( 1 + (-10.9 - 10.9i)T + 61iT^{2} \) |
| 67 | \( 1 + (3.83 - 3.83i)T - 67iT^{2} \) |
| 71 | \( 1 - 9.96iT - 71T^{2} \) |
| 73 | \( 1 - 11.4iT - 73T^{2} \) |
| 79 | \( 1 + 15.4iT - 79T^{2} \) |
| 83 | \( 1 + (-4.23 - 4.23i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.38T + 89T^{2} \) |
| 97 | \( 1 - 6.00T + 97T^{2} \) |
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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
−8.766500377094346772311473364957, −7.69740785506274058235775750355, −7.03771762268476470443486604727, −6.56463122547399585887686442682, −5.64823864374048306247063723272, −5.10221458772964172035450310152, −4.04963927868193937447401650993, −2.77832359094961610399907307820, −2.47381145757755579453121723117, −1.37996366573559987420222748260,
0.70676204526279483141488133553, 1.63526397091265053833431543187, 2.47592907278518345354563831116, 3.68671963943405815565356676645, 4.67549179059771933585553663880, 5.40201988847016592261887509125, 5.81630065811516438592425483547, 6.53224678530705277481455270377, 7.88213154805780274073795718053, 8.254001970578662046444385020407