L(s) = 1 | + 0.746i·3-s + (0.782 − 2.09i)5-s − i·7-s + 2.44·9-s + 5.90·11-s + 3.20i·13-s + (1.56 + 0.584i)15-s − 2.14i·17-s + 3.56·19-s + 0.746·21-s − 3.75i·23-s + (−3.77 − 3.27i)25-s + 4.06i·27-s − 6.61·29-s + 5.79·31-s + ⋯ |
L(s) = 1 | + 0.431i·3-s + (0.350 − 0.936i)5-s − 0.377i·7-s + 0.814·9-s + 1.78·11-s + 0.890i·13-s + (0.403 + 0.151i)15-s − 0.521i·17-s + 0.818·19-s + 0.163·21-s − 0.782i·23-s + (−0.754 − 0.655i)25-s + 0.782i·27-s − 1.22·29-s + 1.04·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.361065129\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.361065129\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.782 + 2.09i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 - 0.746iT - 3T^{2} \) |
| 11 | \( 1 - 5.90T + 11T^{2} \) |
| 13 | \( 1 - 3.20iT - 13T^{2} \) |
| 17 | \( 1 + 2.14iT - 17T^{2} \) |
| 19 | \( 1 - 3.56T + 19T^{2} \) |
| 23 | \( 1 + 3.75iT - 23T^{2} \) |
| 29 | \( 1 + 6.61T + 29T^{2} \) |
| 31 | \( 1 - 5.79T + 31T^{2} \) |
| 37 | \( 1 + 0.623iT - 37T^{2} \) |
| 41 | \( 1 + 5.43T + 41T^{2} \) |
| 43 | \( 1 - 12.6iT - 43T^{2} \) |
| 47 | \( 1 + 4.31iT - 47T^{2} \) |
| 53 | \( 1 - 2.11iT - 53T^{2} \) |
| 59 | \( 1 - 7.01T + 59T^{2} \) |
| 61 | \( 1 + 1.86T + 61T^{2} \) |
| 67 | \( 1 - 6.88iT - 67T^{2} \) |
| 71 | \( 1 - 1.81T + 71T^{2} \) |
| 73 | \( 1 + 6.11iT - 73T^{2} \) |
| 79 | \( 1 + 4.35T + 79T^{2} \) |
| 83 | \( 1 + 13.1iT - 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 9.65iT - 97T^{2} \) |
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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
−9.165871404491821778006448325733, −8.453292782601029938969649731010, −7.31798676522126937298932388858, −6.70132944587473993099486355076, −5.85200173774362331259477922166, −4.64450217938589526365109843050, −4.37633428844990034513023609823, −3.43808372501697298740312488379, −1.80952818000732024529823625946, −1.02609402615512245878091045794,
1.22140794721375206328617280436, 2.10300663019571400571790258334, 3.36353112303205023265904865673, 3.95644330404305971979981916962, 5.32262288357163838795703829416, 6.08874775239248118158393552099, 6.80756953638603867467081385229, 7.34314838386390361884431617381, 8.214048418181442019670667844615, 9.224955874443120906788754059408