| L(s) = 1 | − 2-s + 4-s − 3.46i·5-s + i·7-s − 8-s + 3.46i·10-s + (0.748 − 3.23i)11-s − 2.27i·13-s − i·14-s + 16-s + 1.31·17-s + 1.04i·19-s − 3.46i·20-s + (−0.748 + 3.23i)22-s − 2.56i·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.54i·5-s + 0.377i·7-s − 0.353·8-s + 1.09i·10-s + (0.225 − 0.974i)11-s − 0.630i·13-s − 0.267i·14-s + 0.250·16-s + 0.319·17-s + 0.239i·19-s − 0.774i·20-s + (−0.159 + 0.688i)22-s − 0.534i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9174107825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9174107825\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (-0.748 + 3.23i)T \) |
| good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 13 | \( 1 + 2.27iT - 13T^{2} \) |
| 17 | \( 1 - 1.31T + 17T^{2} \) |
| 19 | \( 1 - 1.04iT - 19T^{2} \) |
| 23 | \( 1 + 2.56iT - 23T^{2} \) |
| 29 | \( 1 - 2.36T + 29T^{2} \) |
| 31 | \( 1 - 2.87T + 31T^{2} \) |
| 37 | \( 1 + 5.87T + 37T^{2} \) |
| 41 | \( 1 + 3.37T + 41T^{2} \) |
| 43 | \( 1 + 4.69iT - 43T^{2} \) |
| 47 | \( 1 + 12.8iT - 47T^{2} \) |
| 53 | \( 1 - 3.57iT - 53T^{2} \) |
| 59 | \( 1 - 4.94iT - 59T^{2} \) |
| 61 | \( 1 - 9.19iT - 61T^{2} \) |
| 67 | \( 1 + 5.35T + 67T^{2} \) |
| 71 | \( 1 - 4.36iT - 71T^{2} \) |
| 73 | \( 1 - 14.2iT - 73T^{2} \) |
| 79 | \( 1 + 16.8iT - 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 6.49iT - 89T^{2} \) |
| 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
−8.934828649193505637623582256335, −8.618801132850672387840291530736, −8.053469222994907833835477237837, −6.92588069195885452323447943930, −5.78976309783419786417805192012, −5.30186981233281456814374158641, −4.13410969008376753849981824516, −2.93975021874866502682509156134, −1.52369296775105161337611988516, −0.48590570655570130461030190108,
1.62345541657682640640424482163, 2.71099599554870684265431744521, 3.63242949509299745149188850276, 4.79094201386444665122420767037, 6.20581163764950295788983692335, 6.79597932019001987298288151185, 7.37143211539060636347468026895, 8.124371908476952895876349378963, 9.341575807963645682856297786291, 9.869736364494581572550299196386
