| L(s) = 1 | + (0.225 + 0.0733i)2-s + (1.21 − 1.67i)3-s + (−1.57 − 1.14i)4-s + (−2.18 − 0.492i)5-s + (0.396 − 0.288i)6-s − i·7-s + (−0.549 − 0.756i)8-s + (−0.392 − 1.20i)9-s + (−0.455 − 0.270i)10-s + (0.360 − 1.10i)11-s + (−3.82 + 1.24i)12-s + (4.17 − 1.35i)13-s + (0.0733 − 0.225i)14-s + (−3.47 + 3.04i)15-s + (1.13 + 3.48i)16-s + (−0.554 − 0.763i)17-s + ⋯ |
| L(s) = 1 | + (0.159 + 0.0518i)2-s + (0.701 − 0.965i)3-s + (−0.786 − 0.571i)4-s + (−0.975 − 0.220i)5-s + (0.161 − 0.117i)6-s − 0.377i·7-s + (−0.194 − 0.267i)8-s + (−0.130 − 0.403i)9-s + (−0.144 − 0.0856i)10-s + (0.108 − 0.334i)11-s + (−1.10 + 0.358i)12-s + (1.15 − 0.376i)13-s + (0.0195 − 0.0602i)14-s + (−0.896 + 0.787i)15-s + (0.283 + 0.871i)16-s + (−0.134 − 0.185i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.717855 - 0.871514i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.717855 - 0.871514i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.18 + 0.492i)T \) |
| 7 | \( 1 + iT \) |
| good | 2 | \( 1 + (-0.225 - 0.0733i)T + (1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (-1.21 + 1.67i)T + (-0.927 - 2.85i)T^{2} \) |
| 11 | \( 1 + (-0.360 + 1.10i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-4.17 + 1.35i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.554 + 0.763i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.0797 - 0.0579i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-3.64 - 1.18i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (3.39 + 2.46i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-6.01 + 4.37i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (4.51 - 1.46i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.64 + 5.07i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 8.35iT - 43T^{2} \) |
| 47 | \( 1 + (2.05 - 2.82i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (5.75 - 7.92i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-4.72 - 14.5i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.06 + 6.35i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-0.383 - 0.527i)T + (-20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (12.9 + 9.37i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-11.0 - 3.57i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (10.0 + 7.32i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (2.50 + 3.45i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.0411 + 0.126i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-10.1 + 13.9i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−12.85618402111031933934658738314, −11.55269583836159384739065010395, −10.51446440671301256778269393859, −9.053004149785115960152954966610, −8.321383440781245577748944739281, −7.43676586482666991018123894219, −6.13869626289943930962516677513, −4.60233183490130266737406903012, −3.36354987585251391419296946533, −1.07628224677438247192895832537,
3.20043320133621328864661869404, 3.93737070890581735346014386186, 4.94047994574051282433140687187, 6.87269881997782164364466502894, 8.409333741379194405922906775978, 8.703300844236255654983233847523, 9.825053613956309844339056030565, 11.03735754455563434570383291676, 12.04683997567962377196462324692, 12.99335219975382076197199405715
