L(s) = 1 | + (1.41 − i)3-s − 4.24i·5-s + i·7-s + (1.00 − 2.82i)9-s + 4.24·11-s − 2·13-s + (−4.24 − 6i)15-s + 7.07i·17-s − 4i·19-s + (1 + 1.41i)21-s + 1.41·23-s − 12.9·25-s + (−1.41 − 5.00i)27-s − 2.82i·29-s − 2i·31-s + ⋯ |
L(s) = 1 | + (0.816 − 0.577i)3-s − 1.89i·5-s + 0.377i·7-s + (0.333 − 0.942i)9-s + 1.27·11-s − 0.554·13-s + (−1.09 − 1.54i)15-s + 1.71i·17-s − 0.917i·19-s + (0.218 + 0.308i)21-s + 0.294·23-s − 2.59·25-s + (−0.272 − 0.962i)27-s − 0.525i·29-s − 0.359i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28447 - 1.52361i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28447 - 1.52361i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.41 + i)T \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 4.24iT - 5T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 7.07iT - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 5.65iT - 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 + 9.89iT - 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.796012052781630190187509904504, −9.195275975483909467784840210835, −8.568021392915641977384319891711, −7.995810697959164256697538490677, −6.74910554981350092535816366830, −5.77761464814791610263279151894, −4.58153993771419100457007259389, −3.77133077181238926323244474587, −2.06505240811029814122507348554, −1.05578848175629554390963341314,
2.15295781254606517118379312516, 3.23365520787058157105723438170, 3.83731975708371451139929277015, 5.19273371515305651271300810021, 6.73012105073072881703946465272, 7.07984132504543770430997395876, 8.038223960623643298242847950683, 9.306160842258568765817756687645, 9.857581637363525441335210536503, 10.62425752592734083047865656216