| L(s) = 1 | + 1.57i·3-s + 0.549i·5-s + 0.517·9-s − 2.39i·11-s − 3.96i·13-s − 0.866·15-s + 4.21·17-s + 6.64i·19-s + 2.34·23-s + 4.69·25-s + 5.54i·27-s − 8.21i·29-s + 0.866·31-s + 3.76·33-s − 0.265i·37-s + ⋯ |
| L(s) = 1 | + 0.909i·3-s + 0.245i·5-s + 0.172·9-s − 0.720i·11-s − 1.10i·13-s − 0.223·15-s + 1.02·17-s + 1.52i·19-s + 0.489·23-s + 0.939·25-s + 1.06i·27-s − 1.52i·29-s + 0.155·31-s + 0.655·33-s − 0.0436i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.875268372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.875268372\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 1.57iT - 3T^{2} \) |
| 5 | \( 1 - 0.549iT - 5T^{2} \) |
| 11 | \( 1 + 2.39iT - 11T^{2} \) |
| 13 | \( 1 + 3.96iT - 13T^{2} \) |
| 17 | \( 1 - 4.21T + 17T^{2} \) |
| 19 | \( 1 - 6.64iT - 19T^{2} \) |
| 23 | \( 1 - 2.34T + 23T^{2} \) |
| 29 | \( 1 + 8.21iT - 29T^{2} \) |
| 31 | \( 1 - 0.866T + 31T^{2} \) |
| 37 | \( 1 + 0.265iT - 37T^{2} \) |
| 41 | \( 1 - 6.24T + 41T^{2} \) |
| 43 | \( 1 + 5.35iT - 43T^{2} \) |
| 47 | \( 1 - 2.59T + 47T^{2} \) |
| 53 | \( 1 - 10.8iT - 53T^{2} \) |
| 59 | \( 1 - 3.77iT - 59T^{2} \) |
| 61 | \( 1 - 7.13iT - 61T^{2} \) |
| 67 | \( 1 - 2.67iT - 67T^{2} \) |
| 71 | \( 1 + 8.76T + 71T^{2} \) |
| 73 | \( 1 + 4.66T + 73T^{2} \) |
| 79 | \( 1 + 0.616T + 79T^{2} \) |
| 83 | \( 1 - 1.09iT - 83T^{2} \) |
| 89 | \( 1 - 6.39T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−9.694101289327696035934182381842, −8.822828120811796146542278751795, −7.957531510752281017941611279479, −7.30773236147984561691475951125, −5.96673287865999365831403881854, −5.55005591011541471608988127052, −4.40611057910497230908842718511, −3.58903015924606180163727973865, −2.80120618868688771897512637700, −1.06717187075386221947111380126,
0.992185203468132920175220695897, 1.96558955011284059471403993728, 3.11103835447292380131832431765, 4.46588215099134614360666220936, 5.06817124669869480211297665706, 6.35443234644650879154433665740, 7.03619901355484989967944762583, 7.45030275206052672305684994046, 8.551849670418406618395011603457, 9.242852851815285373008101501530
