L(s) = 1 | + 0.889·2-s + 2.45i·3-s − 1.20·4-s + 0.645·5-s + 2.18i·6-s + i·7-s − 2.85·8-s − 3.01·9-s + 0.574·10-s + 3.84i·11-s − 2.96i·12-s + 2.91i·13-s + 0.889i·14-s + 1.58i·15-s − 0.121·16-s − 3.75i·17-s + ⋯ |
L(s) = 1 | + 0.628·2-s + 1.41i·3-s − 0.604·4-s + 0.288·5-s + 0.890i·6-s + 0.377i·7-s − 1.00·8-s − 1.00·9-s + 0.181·10-s + 1.15i·11-s − 0.855i·12-s + 0.808i·13-s + 0.237i·14-s + 0.408i·15-s − 0.0303·16-s − 0.910i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.714168 + 1.20257i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.714168 + 1.20257i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - iT \) |
| 41 | \( 1 + (-5.62 + 3.06i)T \) |
good | 2 | \( 1 - 0.889T + 2T^{2} \) |
| 3 | \( 1 - 2.45iT - 3T^{2} \) |
| 5 | \( 1 - 0.645T + 5T^{2} \) |
| 11 | \( 1 - 3.84iT - 11T^{2} \) |
| 13 | \( 1 - 2.91iT - 13T^{2} \) |
| 17 | \( 1 + 3.75iT - 17T^{2} \) |
| 19 | \( 1 + 3.44iT - 19T^{2} \) |
| 23 | \( 1 - 3.35T + 23T^{2} \) |
| 29 | \( 1 - 5.39iT - 29T^{2} \) |
| 31 | \( 1 - 6.35T + 31T^{2} \) |
| 37 | \( 1 - 5.14T + 37T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 6.65iT - 47T^{2} \) |
| 53 | \( 1 + 2.47iT - 53T^{2} \) |
| 59 | \( 1 - 6.30T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 3.57iT - 67T^{2} \) |
| 71 | \( 1 - 8.70iT - 71T^{2} \) |
| 73 | \( 1 + 6.04T + 73T^{2} \) |
| 79 | \( 1 + 6.53iT - 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 1.08iT - 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−12.12871113110360457246668120350, −11.22021886984649448681548999435, −9.983113368458188204451202352737, −9.405246124855350367424230374312, −8.819697435828654529296147840156, −7.10509566192074857787278583624, −5.66920129256487186037793726400, −4.74170130107980534531921607807, −4.20313398631127862755777917570, −2.76750286080905500384408081578,
0.940100185991800980986837497682, 2.83435754709374218739923272395, 4.17991434253598939917359518330, 5.88019796666282969913253317687, 6.12799405611511581328692727752, 7.75546884918820654930438914003, 8.279467526067505198538436368563, 9.547091189489784902940441914548, 10.75754711073382869319642554421, 11.89901171148845996605697053914