| 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
−8.254001970578662046444385020407, −7.88213154805780274073795718053, −6.53224678530705277481455270377, −5.81630065811516438592425483547, −5.40201988847016592261887509125, −4.67549179059771933585553663880, −3.68671963943405815565356676645, −2.47592907278518345354563831116, −1.63526397091265053833431543187, −0.70676204526279483141488133553,
1.37996366573559987420222748260, 2.47381145757755579453121723117, 2.77832359094961610399907307820, 4.04963927868193937447401650993, 5.10221458772964172035450310152, 5.64823864374048306247063723272, 6.56463122547399585887686442682, 7.03771762268476470443486604727, 7.69740785506274058235775750355, 8.766500377094346772311473364957
