| L(s) = 1 | + (2.02 − 1.31i)2-s + (0.521 − 0.200i)3-s + (1.55 − 3.50i)4-s + (−0.881 + 2.05i)5-s + (0.792 − 1.09i)6-s + (−1.40 − 2.23i)7-s + (−0.693 − 4.37i)8-s + (−1.99 + 1.79i)9-s + (0.918 + 5.32i)10-s + (0.978 − 1.08i)11-s + (0.112 − 2.13i)12-s + (2.47 + 4.84i)13-s + (−5.80 − 2.68i)14-s + (−0.0481 + 1.24i)15-s + (−2.03 − 2.25i)16-s + (0.438 − 0.355i)17-s + ⋯ |
| L(s) = 1 | + (1.43 − 0.930i)2-s + (0.300 − 0.115i)3-s + (0.779 − 1.75i)4-s + (−0.394 + 0.919i)5-s + (0.323 − 0.445i)6-s + (−0.532 − 0.846i)7-s + (−0.245 − 1.54i)8-s + (−0.665 + 0.599i)9-s + (0.290 + 1.68i)10-s + (0.294 − 0.327i)11-s + (0.0323 − 0.617i)12-s + (0.685 + 1.34i)13-s + (−1.55 − 0.716i)14-s + (−0.0124 + 0.322i)15-s + (−0.508 − 0.564i)16-s + (0.106 − 0.0861i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.89719 - 1.22957i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.89719 - 1.22957i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.881 - 2.05i)T \) |
| 7 | \( 1 + (1.40 + 2.23i)T \) |
| good | 2 | \( 1 + (-2.02 + 1.31i)T + (0.813 - 1.82i)T^{2} \) |
| 3 | \( 1 + (-0.521 + 0.200i)T + (2.22 - 2.00i)T^{2} \) |
| 11 | \( 1 + (-0.978 + 1.08i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-2.47 - 4.84i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.438 + 0.355i)T + (3.53 - 16.6i)T^{2} \) |
| 19 | \( 1 + (1.75 - 0.779i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (4.99 + 7.69i)T + (-9.35 + 21.0i)T^{2} \) |
| 29 | \( 1 + (-2.15 - 2.96i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (4.12 + 0.433i)T + (30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-7.09 - 0.371i)T + (36.7 + 3.86i)T^{2} \) |
| 41 | \( 1 + (4.45 + 1.44i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-0.695 - 0.695i)T + 43iT^{2} \) |
| 47 | \( 1 + (-7.44 + 9.19i)T + (-9.77 - 45.9i)T^{2} \) |
| 53 | \( 1 + (0.856 + 2.23i)T + (-39.3 + 35.4i)T^{2} \) |
| 59 | \( 1 + (0.998 - 0.212i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-2.23 + 10.5i)T + (-55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (-3.37 - 4.16i)T + (-13.9 + 65.5i)T^{2} \) |
| 71 | \( 1 + (3.85 - 2.80i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.320 - 6.11i)T + (-72.6 + 7.63i)T^{2} \) |
| 79 | \( 1 + (1.01 - 0.106i)T + (77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (9.51 - 1.50i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (8.87 + 1.88i)T + (81.3 + 36.1i)T^{2} \) |
| 97 | \( 1 + (-7.91 - 1.25i)T + (92.2 + 29.9i)T^{2} \) |
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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
−12.59525258576304008582671955042, −11.52010538388349524593808413399, −10.93672193980486461628988496357, −10.10648455361768938077662857094, −8.444473348286529734531496188101, −6.89423840950091535985220060574, −6.05011617241894587046076831410, −4.31804327501472567733060575297, −3.53044878309069656377542211185, −2.27760312706711288495487676139,
3.12230308024321073590455859600, 4.12292137943475119889124409372, 5.58945933407649075455056532909, 6.03695759659953026495394114830, 7.65409083444097168975977367981, 8.529362962337696004044944020136, 9.592640584946864232701870542893, 11.54824689839762935299770030446, 12.31697756014277440679775369037, 12.97666932293037767096921410189
