L(s) = 1 | + (1.38 − 0.285i)2-s − 3-s + (1.83 − 0.791i)4-s + 2.80·5-s + (−1.38 + 0.285i)6-s − 0.415·7-s + (2.31 − 1.62i)8-s + 9-s + (3.88 − 0.801i)10-s + 5.54i·11-s + (−1.83 + 0.791i)12-s − 3.19i·13-s + (−0.575 + 0.118i)14-s − 2.80·15-s + (2.74 − 2.90i)16-s + 2.01i·17-s + ⋯ |
L(s) = 1 | + (0.979 − 0.201i)2-s − 0.577·3-s + (0.918 − 0.395i)4-s + 1.25·5-s + (−0.565 + 0.116i)6-s − 0.156·7-s + (0.819 − 0.572i)8-s + 0.333·9-s + (1.22 − 0.253i)10-s + 1.67i·11-s + (−0.530 + 0.228i)12-s − 0.884i·13-s + (−0.153 + 0.0316i)14-s − 0.724·15-s + (0.687 − 0.726i)16-s + 0.488i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.65132 - 0.424856i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.65132 - 0.424856i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.38 + 0.285i)T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + (2.87 + 3.83i)T \) |
good | 5 | \( 1 - 2.80T + 5T^{2} \) |
| 7 | \( 1 + 0.415T + 7T^{2} \) |
| 11 | \( 1 - 5.54iT - 11T^{2} \) |
| 13 | \( 1 + 3.19iT - 13T^{2} \) |
| 17 | \( 1 - 2.01iT - 17T^{2} \) |
| 19 | \( 1 + 2.50iT - 19T^{2} \) |
| 29 | \( 1 + 4.69iT - 29T^{2} \) |
| 31 | \( 1 - 8.16iT - 31T^{2} \) |
| 37 | \( 1 - 3.59T + 37T^{2} \) |
| 41 | \( 1 - 2.52T + 41T^{2} \) |
| 43 | \( 1 + 1.80iT - 43T^{2} \) |
| 47 | \( 1 - 3.58iT - 47T^{2} \) |
| 53 | \( 1 + 0.270T + 53T^{2} \) |
| 59 | \( 1 + 0.657T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 5.41iT - 67T^{2} \) |
| 71 | \( 1 - 6.09iT - 71T^{2} \) |
| 73 | \( 1 + 2.82T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + 11.1iT - 83T^{2} \) |
| 89 | \( 1 - 4.02iT - 89T^{2} \) |
| 97 | \( 1 - 14.5iT - 97T^{2} \) |
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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
−10.58365731549925744841331282297, −10.19694999171884814524845843140, −9.400779502803125333718362270896, −7.74651778786891191668729473695, −6.69873664035145405870556416197, −6.04559023980213270175086636608, −5.13117873614760478849917771683, −4.33161775376974616171635727749, −2.70332499154129990748555199660, −1.65027730466867523352848942889,
1.68628706286570737655071679087, 3.06596032846645680473231165797, 4.29861977735616516643843621095, 5.64846940391564463960606133521, 5.88506459749506433542958950204, 6.78065039424603304215911803689, 7.964008933428636471330341416877, 9.201207356628444107174987839530, 10.09276445822091843643664167696, 11.14896198678478527302525197264