L(s) = 1 | − 5.77i·2-s − 5.19·3-s − 17.3·4-s + 34.3·5-s + 29.9i·6-s − 39.2·7-s + 7.56i·8-s + 27·9-s − 198. i·10-s + 56.8i·11-s + 89.9·12-s − 232. i·13-s + 226. i·14-s − 178.·15-s − 233.·16-s − 391.·17-s + ⋯ |
L(s) = 1 | − 1.44i·2-s − 0.577·3-s − 1.08·4-s + 1.37·5-s + 0.833i·6-s − 0.801·7-s + 0.118i·8-s + 0.333·9-s − 1.98i·10-s + 0.469i·11-s + 0.624·12-s − 1.37i·13-s + 1.15i·14-s − 0.793·15-s − 0.911·16-s − 1.35·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.7233871187\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7233871187\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 5.19T \) |
| 59 | \( 1 + (1.54e3 + 3.11e3i)T \) |
good | 2 | \( 1 + 5.77iT - 16T^{2} \) |
| 5 | \( 1 - 34.3T + 625T^{2} \) |
| 7 | \( 1 + 39.2T + 2.40e3T^{2} \) |
| 11 | \( 1 - 56.8iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 232. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 391.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 352.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 173. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 90.7T + 7.07e5T^{2} \) |
| 31 | \( 1 - 128. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 413. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 761.T + 2.82e6T^{2} \) |
| 43 | \( 1 - 874. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 895. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 3.68e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 1.49e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 3.68e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 9.23e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 4.30e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 4.70e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 398. iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.24e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 397. iT - 8.85e7T^{2} \) |
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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
−11.15168578423764386268585431099, −10.33849020163951389465504570169, −9.865658976802396617283012326493, −8.829095356087646481737814308723, −6.80601899447201424379164640093, −5.88155886486416432521079919180, −4.49844984105966748980079289940, −2.89555870759909995501286262742, −1.83589294007216688322623115807, −0.26384579991923362018719698794,
2.10639894315086816830803794966, 4.43813580620703121690637557348, 5.69433877806528254175504649559, 6.42593780388775549528789396330, 6.93531734894113324355575057209, 8.695613876017793615280775255717, 9.361916374556345192711423869379, 10.52085902288214753384404352255, 11.69629186033938691394061501288, 13.25189842873369345363212690616