| L(s) = 1 | + (0.707 − 0.707i)3-s + (0.707 − 0.707i)5-s + (0.707 + 0.707i)7-s − i·9-s + (0.707 + 0.707i)11-s + 13-s − i·15-s + i·19-s + 21-s + (−0.707 − 0.707i)23-s − i·25-s + (−0.707 − 0.707i)27-s + (0.707 − 0.707i)29-s + (−0.707 + 0.707i)31-s + 33-s + ⋯ |
| L(s) = 1 | + (0.707 − 0.707i)3-s + (0.707 − 0.707i)5-s + (0.707 + 0.707i)7-s − i·9-s + (0.707 + 0.707i)11-s + 13-s − i·15-s + i·19-s + 21-s + (−0.707 − 0.707i)23-s − i·25-s + (−0.707 − 0.707i)27-s + (0.707 − 0.707i)29-s + (−0.707 + 0.707i)31-s + 33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.669265358 - 1.179122914i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.669265358 - 1.179122914i\) |
| \(L(1)\) |
\(\approx\) |
\(1.662160294 - 0.4587835224i\) |
| \(L(1)\) |
\(\approx\) |
\(1.662160294 - 0.4587835224i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
| 11 | \( 1 + (0.707 + 0.707i)T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 + iT \) |
| 23 | \( 1 + (-0.707 - 0.707i)T \) |
| 29 | \( 1 + (0.707 - 0.707i)T \) |
| 31 | \( 1 + (-0.707 + 0.707i)T \) |
| 37 | \( 1 + (-0.707 + 0.707i)T \) |
| 41 | \( 1 + (-0.707 - 0.707i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - iT \) |
| 59 | \( 1 - iT \) |
| 61 | \( 1 + (0.707 + 0.707i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.707 + 0.707i)T \) |
| 73 | \( 1 + (-0.707 + 0.707i)T \) |
| 79 | \( 1 + (-0.707 - 0.707i)T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (-0.707 + 0.707i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−28.1608552639307353112749117327, −27.25101266192710571014437284409, −26.3706172449142575499959941173, −25.68018359598218420384276853149, −24.625580245416464556382722122598, −23.41059294372573625980032088949, −22.046583887792558917249939491162, −21.48951641448506791987317255855, −20.454369562779362927301977348623, −19.5113194577399281085351014856, −18.26687401019073542485365491980, −17.226289150641326156759882666415, −16.09632477580884375411954259621, −14.89856957167039080279280084668, −13.97551409178483791742230513929, −13.487516814161261679734395627920, −11.28170230013022208903248869287, −10.66083981237477988263044152836, −9.45460183653809494759045229611, −8.42953955540193081917570339177, −7.10194932397640497929092507481, −5.69102491716446271579504687629, −4.18918306918031502827655191671, −3.11434229598499777851811538929, −1.56749574318535922667323069757,
1.338762378072862996473081736677, 2.16868677976124885573125350111, 3.98546087836715496273854210359, 5.54901031154850531924604184634, 6.67304038169637376179758031102, 8.271393165835573376466468191101, 8.81714356520994176217450918263, 10.05740959132042946186650194560, 11.89203518878633938198976004895, 12.56218934419455765338040897938, 13.80767671029910544394730027579, 14.51810372655565449334950178424, 15.771625657256662965553296646294, 17.24240887169758443884402937552, 18.05856965124848448227460181115, 18.943001553241928450339778922942, 20.40563116004294238847244823003, 20.75350168276141973331359337864, 22.00612668100655537625902714674, 23.43556385788020978342878051250, 24.43478706864757182803498894309, 25.18273631739654486068795739009, 25.70619217199141240025678786908, 27.23346627899970943161027803412, 28.25631034621621135471886796876
