L(s) = 1 | + (0.312 + 1.37i)2-s + (−0.599 + 0.599i)3-s + (−1.80 + 0.862i)4-s + (0.974 + 0.974i)5-s + (−1.01 − 0.640i)6-s + i·7-s + (−1.75 − 2.21i)8-s + 2.28i·9-s + (−1.04 + 1.64i)10-s + (−1.72 − 1.72i)11-s + (0.565 − 1.59i)12-s + (1.90 − 1.90i)13-s + (−1.37 + 0.312i)14-s − 1.16·15-s + (2.51 − 3.11i)16-s + 6.71·17-s + ⋯ |
L(s) = 1 | + (0.220 + 0.975i)2-s + (−0.346 + 0.346i)3-s + (−0.902 + 0.431i)4-s + (0.436 + 0.436i)5-s + (−0.414 − 0.261i)6-s + 0.377i·7-s + (−0.619 − 0.784i)8-s + 0.760i·9-s + (−0.328 + 0.521i)10-s + (−0.519 − 0.519i)11-s + (0.163 − 0.461i)12-s + (0.528 − 0.528i)13-s + (−0.368 + 0.0835i)14-s − 0.302·15-s + (0.628 − 0.777i)16-s + 1.62·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.514127 + 0.865287i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.514127 + 0.865287i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.312 - 1.37i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + (0.599 - 0.599i)T - 3iT^{2} \) |
| 5 | \( 1 + (-0.974 - 0.974i)T + 5iT^{2} \) |
| 11 | \( 1 + (1.72 + 1.72i)T + 11iT^{2} \) |
| 13 | \( 1 + (-1.90 + 1.90i)T - 13iT^{2} \) |
| 17 | \( 1 - 6.71T + 17T^{2} \) |
| 19 | \( 1 + (-2.94 + 2.94i)T - 19iT^{2} \) |
| 23 | \( 1 - 5.29iT - 23T^{2} \) |
| 29 | \( 1 + (3.03 - 3.03i)T - 29iT^{2} \) |
| 31 | \( 1 - 1.19T + 31T^{2} \) |
| 37 | \( 1 + (2.25 + 2.25i)T + 37iT^{2} \) |
| 41 | \( 1 + 3.94iT - 41T^{2} \) |
| 43 | \( 1 + (7.02 + 7.02i)T + 43iT^{2} \) |
| 47 | \( 1 + 3.06T + 47T^{2} \) |
| 53 | \( 1 + (3.01 + 3.01i)T + 53iT^{2} \) |
| 59 | \( 1 + (-4.96 - 4.96i)T + 59iT^{2} \) |
| 61 | \( 1 + (9.69 - 9.69i)T - 61iT^{2} \) |
| 67 | \( 1 + (-3.55 + 3.55i)T - 67iT^{2} \) |
| 71 | \( 1 - 11.5iT - 71T^{2} \) |
| 73 | \( 1 + 10.3iT - 73T^{2} \) |
| 79 | \( 1 - 4.06T + 79T^{2} \) |
| 83 | \( 1 + (-9.17 + 9.17i)T - 83iT^{2} \) |
| 89 | \( 1 - 16.9iT - 89T^{2} \) |
| 97 | \( 1 + 2.51T + 97T^{2} \) |
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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
−13.94081492807806634527451538930, −13.35112797097382959337878238806, −12.01314013816505534228764001605, −10.66974691256189641497151014175, −9.708462844606809467609867149217, −8.346041957032577085710283277250, −7.34349757436623592798036630191, −5.77697423238169805677788028236, −5.24536839764671958185821472954, −3.30246262265829076682312758205,
1.36878422182036215515004795420, 3.48511268333851522740498253336, 5.04407125882127847358829854700, 6.23641950381196875428641242648, 7.957357060723954278198668712356, 9.414559893256189628295654595664, 10.10184832434347206319185506891, 11.40874725828124604153034763689, 12.30505095412263722214208532600, 13.00410807527890862571876079057