L(s) = 1 | + (−0.984 − 0.173i)2-s + (−1.07 + 1.28i)3-s + (0.939 + 0.342i)4-s + (1.28 − 1.07i)6-s + (0.411 − 0.237i)7-s + (−0.866 − 0.5i)8-s + (0.0320 + 0.181i)9-s + (2.79 − 4.84i)11-s + (−1.45 + 0.838i)12-s + (−1.58 − 1.88i)13-s + (−0.446 + 0.162i)14-s + (0.766 + 0.642i)16-s + (−2.82 − 0.498i)17-s − 0.184i·18-s + (−3.17 + 2.98i)19-s + ⋯ |
L(s) = 1 | + (−0.696 − 0.122i)2-s + (−0.622 + 0.742i)3-s + (0.469 + 0.171i)4-s + (0.524 − 0.440i)6-s + (0.155 − 0.0898i)7-s + (−0.306 − 0.176i)8-s + (0.0106 + 0.0606i)9-s + (0.843 − 1.46i)11-s + (−0.419 + 0.242i)12-s + (−0.438 − 0.523i)13-s + (−0.119 + 0.0434i)14-s + (0.191 + 0.160i)16-s + (−0.686 − 0.121i)17-s − 0.0435i·18-s + (−0.729 + 0.683i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0990 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0990 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.396291 - 0.358819i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.396291 - 0.358819i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.984 + 0.173i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (3.17 - 2.98i)T \) |
good | 3 | \( 1 + (1.07 - 1.28i)T + (-0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (-0.411 + 0.237i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.79 + 4.84i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.58 + 1.88i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.82 + 0.498i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (1.50 - 4.12i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (1.51 + 8.59i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.88 + 4.99i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.02iT - 37T^{2} \) |
| 41 | \( 1 + (-5.18 - 4.34i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.157 - 0.433i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-2.59 + 0.456i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-2.70 + 7.43i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.42 + 8.06i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.98 + 1.08i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.148 - 0.0262i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (4.48 - 1.63i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-5.91 + 7.04i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (9.48 + 7.96i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-11.4 + 6.59i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.23 + 1.03i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-4.95 - 0.873i)T + (91.1 + 33.1i)T^{2} \) |
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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
−9.831883187792758741931976420783, −9.220067801683725399722950129696, −8.213465938188960757988853050202, −7.56124766709260070546806548792, −6.16820522484552986072693149688, −5.75063030906399842430040443107, −4.42074986269591047464706543566, −3.56572488235519788785062969470, −2.08159492460153901331935598825, −0.35035019014196117243999180526,
1.36580522735743447579897334037, 2.31421487615407029570891977711, 4.11137740333772110947101583200, 5.10131608800797567110250200785, 6.41889898202277667280650130933, 6.86168379384812909857984304249, 7.42354051870712784531740019628, 8.785110210568338070576011123973, 9.199545647727353838401111610025, 10.24582390421168989457498601408