| L(s) = 1 | − 2-s − i·3-s + 4-s + i·6-s + (−1.87 − 1.87i)7-s − 8-s − 9-s − 3.74i·11-s − i·12-s + 2i·13-s + (1.87 + 1.87i)14-s + 16-s + 3.74·17-s + 18-s + (−1.87 + 1.87i)21-s + 3.74i·22-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577i·3-s + 0.5·4-s + 0.408i·6-s + (−0.707 − 0.707i)7-s − 0.353·8-s − 0.333·9-s − 1.12i·11-s − 0.288i·12-s + 0.554i·13-s + (0.499 + 0.499i)14-s + 0.250·16-s + 0.907·17-s + 0.235·18-s + (−0.408 + 0.408i)21-s + 0.797i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.109i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0211038 + 0.384824i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0211038 + 0.384824i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 7 | \( 1 + (1.87 + 1.87i)T \) |
| 23 | \( 1 + (3 + 3.74i)T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 3.74iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 3.74T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 - 3.74iT - 37T^{2} \) |
| 41 | \( 1 - 2iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 7.48iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 3.74iT - 79T^{2} \) |
| 83 | \( 1 + 3.74T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.621527516239330690069185582069, −8.678572046111310354811166346288, −7.936558670224995244164469294498, −7.13395284651881474799843090172, −6.34474618527743319075033249532, −5.58563532061931022759432550801, −3.93240019432148899056232443315, −3.01871155424124785698321771280, −1.58173653741041645162213418788, −0.21838452420015866643731914774,
1.90231437888449660526363786799, 3.07534140117014699355938283479, 4.11536461193976544068174352103, 5.52351381928170091450649863454, 6.02180545433059201664409736172, 7.38330976588463643333262110134, 7.88723194993838230313103595048, 9.135223869314062248203972534706, 9.585316591900208149758700905995, 10.13733347425257950068428351057
