| L(s) = 1 | + 1.58i·5-s + (0.618 + 2.57i)7-s − 1.61·11-s + 3.38·13-s + 2.57i·17-s + (−1.29 + 4.16i)19-s − 7.47·23-s + 2.47·25-s + 8.71i·29-s − 8.06·31-s + (−4.09 + 0.982i)35-s − 5.38i·37-s + 8.06·41-s − 5.70·43-s − 4.76i·47-s + ⋯ |
| L(s) = 1 | + 0.711i·5-s + (0.233 + 0.972i)7-s − 0.487·11-s + 0.939·13-s + 0.623i·17-s + (−0.296 + 0.954i)19-s − 1.55·23-s + 0.494·25-s + 1.61i·29-s − 1.44·31-s + (−0.691 + 0.166i)35-s − 0.885i·37-s + 1.25·41-s − 0.870·43-s − 0.695i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.026490598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.026490598\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.618 - 2.57i)T \) |
| 19 | \( 1 + (1.29 - 4.16i)T \) |
| good | 5 | \( 1 - 1.58iT - 5T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 - 3.38T + 13T^{2} \) |
| 17 | \( 1 - 2.57iT - 17T^{2} \) |
| 23 | \( 1 + 7.47T + 23T^{2} \) |
| 29 | \( 1 - 8.71iT - 29T^{2} \) |
| 31 | \( 1 + 8.06T + 31T^{2} \) |
| 37 | \( 1 + 5.38iT - 37T^{2} \) |
| 41 | \( 1 - 8.06T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 + 4.76iT - 47T^{2} \) |
| 53 | \( 1 + 14.0iT - 53T^{2} \) |
| 59 | \( 1 + 0.799T + 59T^{2} \) |
| 61 | \( 1 - 6.73iT - 61T^{2} \) |
| 67 | \( 1 + 2.05iT - 67T^{2} \) |
| 71 | \( 1 - 12.0iT - 71T^{2} \) |
| 73 | \( 1 + 9.30iT - 73T^{2} \) |
| 79 | \( 1 - 10.7iT - 79T^{2} \) |
| 83 | \( 1 + 2.94iT - 83T^{2} \) |
| 89 | \( 1 - 8.37T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−8.532487966313609442604740789197, −8.048621440367089008238112666359, −7.18562782552944775968411897293, −6.36016346634099301421136557927, −5.76476246758700158283968935850, −5.18384638662739013433498689293, −3.92628105030431241802483463539, −3.39859330235371063147980647975, −2.31055602034985936782372870263, −1.62239925364035530825913660853,
0.27844887143673088651115118446, 1.25168890381352421813605677205, 2.34800203479382137558233726750, 3.46189540712130906559859172605, 4.33199453649359158257995877567, 4.77206700408021423881445280411, 5.78494969188700799070515326874, 6.41743371465637730950104282349, 7.39735134761185402760823617102, 7.88778845291661389928392401598
