| L(s) = 1 | + (0.203 − 1.72i)3-s − 3.44·5-s + i·7-s + (−2.91 − 0.699i)9-s + 5.42i·13-s + (−0.699 + 5.91i)15-s + 3.15i·17-s + 4.40·19-s + (1.72 + 0.203i)21-s + 4.55·23-s + 6.83·25-s + (−1.79 + 4.87i)27-s − 4.08·29-s − 3.44i·35-s + 3.18i·37-s + ⋯ |
| L(s) = 1 | + (0.117 − 0.993i)3-s − 1.53·5-s + 0.377i·7-s + (−0.972 − 0.233i)9-s + 1.50i·13-s + (−0.180 + 1.52i)15-s + 0.765i·17-s + 1.01·19-s + (0.375 + 0.0443i)21-s + 0.949·23-s + 1.36·25-s + (−0.345 + 0.938i)27-s − 0.757·29-s − 0.581i·35-s + 0.523i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.574156 + 0.394139i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.574156 + 0.394139i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.203 + 1.72i)T \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 + 3.44T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 5.42iT - 13T^{2} \) |
| 17 | \( 1 - 3.15iT - 17T^{2} \) |
| 19 | \( 1 - 4.40T + 19T^{2} \) |
| 23 | \( 1 - 4.55T + 23T^{2} \) |
| 29 | \( 1 + 4.08T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 3.18iT - 37T^{2} \) |
| 41 | \( 1 - 5.95iT - 41T^{2} \) |
| 43 | \( 1 + 9.83T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 6.31T + 53T^{2} \) |
| 59 | \( 1 + 0.641iT - 59T^{2} \) |
| 61 | \( 1 - 10.2iT - 61T^{2} \) |
| 67 | \( 1 - 9.02T + 67T^{2} \) |
| 71 | \( 1 + 3.15T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 1.62iT - 79T^{2} \) |
| 83 | \( 1 - 12.5iT - 83T^{2} \) |
| 89 | \( 1 - 7.80iT - 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−11.17881646353564070564290783587, −9.611671093197260158214742126855, −8.641583620941204715528227287607, −8.065275660770916832811092922513, −7.14672712462825707311455164399, −6.57295705518492788237673829584, −5.22245274060236811464174340290, −4.01599585962053166758974292494, −3.01338555430615761330042736674, −1.47514841662586408117399565735,
0.38624735720477021309646381439, 3.10660813893704458908132572818, 3.59743103504921757231737517813, 4.76268426099529278303626321101, 5.46097789414369192434868379689, 7.08769141266271804656948496810, 7.81855628441496067673722130744, 8.530725098696728093641752996137, 9.553239942595632311120630635560, 10.40106988725070398642618259445
