| L(s) = 1 | + (−0.587 − 0.809i)2-s + (−1.57 − 0.513i)3-s + (−0.309 + 0.951i)4-s + (−0.806 − 2.08i)5-s + (0.513 + 1.57i)6-s + i·7-s + (0.951 − 0.309i)8-s + (−0.195 − 0.142i)9-s + (−1.21 + 1.87i)10-s + (−3.99 + 2.90i)11-s + (0.976 − 1.34i)12-s + (0.873 − 1.20i)13-s + (0.809 − 0.587i)14-s + (0.204 + 3.70i)15-s + (−0.809 − 0.587i)16-s + (1.14 − 0.373i)17-s + ⋯ |
| L(s) = 1 | + (−0.415 − 0.572i)2-s + (−0.911 − 0.296i)3-s + (−0.154 + 0.475i)4-s + (−0.360 − 0.932i)5-s + (0.209 + 0.644i)6-s + 0.377i·7-s + (0.336 − 0.109i)8-s + (−0.0653 − 0.0474i)9-s + (−0.383 + 0.594i)10-s + (−1.20 + 0.876i)11-s + (0.281 − 0.387i)12-s + (0.242 − 0.333i)13-s + (0.216 − 0.157i)14-s + (0.0527 + 0.957i)15-s + (−0.202 − 0.146i)16-s + (0.278 − 0.0906i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0783 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0783 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.129155 + 0.119404i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.129155 + 0.119404i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.587 + 0.809i)T \) |
| 5 | \( 1 + (0.806 + 2.08i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + (1.57 + 0.513i)T + (2.42 + 1.76i)T^{2} \) |
| 11 | \( 1 + (3.99 - 2.90i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.873 + 1.20i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.14 + 0.373i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.07 - 6.37i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (1.85 + 2.55i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (2.66 - 8.20i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.26 - 6.98i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (2.81 - 3.87i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (7.71 + 5.60i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 11.6iT - 43T^{2} \) |
| 47 | \( 1 + (-3.57 - 1.16i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (6.68 + 2.17i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (3.88 + 2.82i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.316 + 0.229i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (9.66 - 3.13i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.17 + 9.77i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.717 + 0.988i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (3.69 - 11.3i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (15.7 - 5.12i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-11.6 + 8.45i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (2.00 + 0.651i)T + (78.4 + 57.0i)T^{2} \) |
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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
−12.08015863006309433550045757055, −10.73515092217291104989203235382, −10.11895348343669727490391734054, −8.876159748317613327097553875625, −8.120851650062596788807608499430, −7.08721342853556307713404158794, −5.58209817836203756146113392772, −4.97656981522537882588526753583, −3.38196142464160317747300078565, −1.57099590610789945065530692044,
0.15591911099816184333085280005, 2.85119780156237027500185061599, 4.44365529433637297657255258860, 5.64140644988749803793085933029, 6.34163441197302428989590267324, 7.51547245004254852698581091354, 8.168607363469267650350140873577, 9.611395625458988209759424887352, 10.45806715480905608652777091989, 11.23681585937376967214144009282
