| L(s) = 1 | + (0.343 + 0.131i)2-s + (2.87 − 0.150i)3-s + (−1.38 − 1.24i)4-s + (−0.280 − 2.21i)5-s + (1.00 + 0.327i)6-s + (−0.714 + 2.54i)7-s + (−0.645 − 1.26i)8-s + (5.27 − 0.554i)9-s + (0.195 − 0.798i)10-s + (−0.0862 + 0.820i)11-s + (−4.17 − 3.38i)12-s + (−0.749 − 0.118i)13-s + (−0.581 + 0.780i)14-s + (−1.14 − 6.34i)15-s + (0.335 + 3.18i)16-s + (−3.24 + 4.98i)17-s + ⋯ |
| L(s) = 1 | + (0.242 + 0.0932i)2-s + (1.66 − 0.0870i)3-s + (−0.692 − 0.623i)4-s + (−0.125 − 0.992i)5-s + (0.411 + 0.133i)6-s + (−0.269 + 0.962i)7-s + (−0.228 − 0.447i)8-s + (1.75 − 0.184i)9-s + (0.0619 − 0.252i)10-s + (−0.0260 + 0.247i)11-s + (−1.20 − 0.976i)12-s + (−0.207 − 0.0329i)13-s + (−0.155 + 0.208i)14-s + (−0.295 − 1.63i)15-s + (0.0837 + 0.797i)16-s + (−0.785 + 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 + 0.464i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.67610 - 0.413057i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67610 - 0.413057i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.280 + 2.21i)T \) |
| 7 | \( 1 + (0.714 - 2.54i)T \) |
| good | 2 | \( 1 + (-0.343 - 0.131i)T + (1.48 + 1.33i)T^{2} \) |
| 3 | \( 1 + (-2.87 + 0.150i)T + (2.98 - 0.313i)T^{2} \) |
| 11 | \( 1 + (0.0862 - 0.820i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (0.749 + 0.118i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (3.24 - 4.98i)T + (-6.91 - 15.5i)T^{2} \) |
| 19 | \( 1 + (-3.66 - 4.06i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-1.40 + 3.65i)T + (-17.0 - 15.3i)T^{2} \) |
| 29 | \( 1 + (-3.19 + 1.03i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.50 + 7.06i)T + (-28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (1.90 - 2.35i)T + (-7.69 - 36.1i)T^{2} \) |
| 41 | \( 1 + (2.45 + 3.38i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (5.78 - 5.78i)T - 43iT^{2} \) |
| 47 | \( 1 + (-8.57 + 5.56i)T + (19.1 - 42.9i)T^{2} \) |
| 53 | \( 1 + (0.0303 + 0.579i)T + (-52.7 + 5.54i)T^{2} \) |
| 59 | \( 1 + (6.58 + 2.93i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (4.93 + 11.0i)T + (-40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (4.87 + 3.16i)T + (27.2 + 61.2i)T^{2} \) |
| 71 | \( 1 + (-1.49 - 4.61i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (7.59 - 6.15i)T + (15.1 - 71.4i)T^{2} \) |
| 79 | \( 1 + (0.0719 - 0.338i)T + (-72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-14.9 + 7.61i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (-10.2 + 4.55i)T + (59.5 - 66.1i)T^{2} \) |
| 97 | \( 1 + (-14.5 - 7.41i)T + (57.0 + 78.4i)T^{2} \) |
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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
−12.95027037228615280541760461032, −12.14699328559508462043728768038, −10.15641854525750595510826180604, −9.306000367571747283752257289532, −8.702603456239689588147060021372, −7.944930708604086032071402644329, −6.14613003903327784011236662455, −4.80480225910697795771045544543, −3.65902971749672014291496630914, −1.92419628692540275800707620796,
2.84241500611380949112971405899, 3.44267498315903156276431115511, 4.66455194984473480540816035440, 7.10506955770338251998097076408, 7.53425158577486428095827678641, 8.835556132708084860241710827396, 9.515529270474293750102508575772, 10.65868245015114293168505681048, 11.93580027321846990007411869864, 13.49305138377679194597894388454
