| L(s) = 1 | + (−2.04 − 0.912i)2-s + (−0.0396 + 0.377i)3-s + (2.02 + 2.25i)4-s + 0.301·5-s + (0.425 − 0.736i)6-s + (2.74 + 3.04i)7-s + (−0.712 − 2.19i)8-s + (2.79 + 0.593i)9-s + (−0.618 − 0.275i)10-s + (−5.66 + 1.20i)11-s + (−0.929 + 0.675i)12-s + (−3.59 + 0.277i)13-s + (−2.84 − 8.74i)14-s + (−0.0119 + 0.113i)15-s + (0.0926 − 0.881i)16-s + (−0.851 − 0.180i)17-s + ⋯ |
| L(s) = 1 | + (−1.44 − 0.644i)2-s + (−0.0228 + 0.217i)3-s + (1.01 + 1.12i)4-s + 0.134·5-s + (0.173 − 0.300i)6-s + (1.03 + 1.15i)7-s + (−0.251 − 0.775i)8-s + (0.931 + 0.197i)9-s + (−0.195 − 0.0870i)10-s + (−1.70 + 0.362i)11-s + (−0.268 + 0.194i)12-s + (−0.997 + 0.0768i)13-s + (−0.759 − 2.33i)14-s + (−0.00309 + 0.0294i)15-s + (0.0231 − 0.220i)16-s + (−0.206 − 0.0438i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.227 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.431971 + 0.342687i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.431971 + 0.342687i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (3.59 - 0.277i)T \) |
| 31 | \( 1 + (-4.27 - 3.56i)T \) |
| good | 2 | \( 1 + (2.04 + 0.912i)T + (1.33 + 1.48i)T^{2} \) |
| 3 | \( 1 + (0.0396 - 0.377i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 - 0.301T + 5T^{2} \) |
| 7 | \( 1 + (-2.74 - 3.04i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (5.66 - 1.20i)T + (10.0 - 4.47i)T^{2} \) |
| 17 | \( 1 + (0.851 + 0.180i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.349 - 3.32i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.161 - 0.178i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (5.06 + 2.25i)T + (19.4 + 21.5i)T^{2} \) |
| 37 | \( 1 + (-2.95 - 5.11i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.88 + 0.837i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (0.344 + 3.27i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (-10.3 - 7.50i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.88 - 5.80i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.470 + 0.209i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-3.67 + 6.36i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.308 + 0.533i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (10.8 + 2.30i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (3.09 - 9.51i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-1.59 - 4.92i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-11.4 + 8.32i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (8.50 - 1.80i)T + (81.3 - 36.1i)T^{2} \) |
| 97 | \( 1 + (2.31 + 2.57i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−11.21977381826568445181397855851, −10.25191148245459224987257423982, −9.860316923828519650252237844569, −8.830450771795397977739582682822, −7.86811989489232180223005404470, −7.47946101293970630846174719912, −5.56512584952317736428423481654, −4.68138971275201588920093410454, −2.57889546208579456808891057814, −1.83635216162961684776859519895,
0.56483174523075723769346411239, 2.10151156654170658311552272718, 4.28682398039900267120694000964, 5.46945335796191501900597008230, 6.92017877078922074113244299676, 7.62195557493235833278798825368, 7.935980525219968415093926482666, 9.211904786379403103979961385541, 10.22525073163178990348208276297, 10.54097442606778963756075835558
