L(s) = 1 | + (0.790 − 1.17i)2-s + (−0.573 + 0.331i)3-s + (−0.750 − 1.85i)4-s + (−0.0650 + 0.934i)6-s + (2.03 + 1.68i)7-s + (−2.76 − 0.585i)8-s + (−1.28 + 2.21i)9-s + (3.12 − 1.80i)11-s + (1.04 + 0.815i)12-s + 5.83·13-s + (3.58 − 1.05i)14-s + (−2.87 + 2.78i)16-s + (0.684 + 1.18i)17-s + (1.58 + 3.25i)18-s + (2.04 − 3.54i)19-s + ⋯ |
L(s) = 1 | + (0.558 − 0.829i)2-s + (−0.331 + 0.191i)3-s + (−0.375 − 0.926i)4-s + (−0.0265 + 0.381i)6-s + (0.770 + 0.637i)7-s + (−0.978 − 0.207i)8-s + (−0.426 + 0.739i)9-s + (0.943 − 0.544i)11-s + (0.301 + 0.235i)12-s + 1.61·13-s + (0.959 − 0.282i)14-s + (−0.718 + 0.695i)16-s + (0.165 + 0.287i)17-s + (0.374 + 0.767i)18-s + (0.469 − 0.813i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.72143 - 0.959999i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.72143 - 0.959999i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.790 + 1.17i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.03 - 1.68i)T \) |
good | 3 | \( 1 + (0.573 - 0.331i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-3.12 + 1.80i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 5.83T + 13T^{2} \) |
| 17 | \( 1 + (-0.684 - 1.18i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.04 + 3.54i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.62 + 2.81i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 5.19T + 29T^{2} \) |
| 31 | \( 1 + (4.43 + 7.67i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-9.34 - 5.39i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 0.832iT - 41T^{2} \) |
| 43 | \( 1 - 3.10T + 43T^{2} \) |
| 47 | \( 1 + (-5.97 - 3.44i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.42 - 3.70i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.73 + 6.47i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.28 - 0.742i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.26 + 2.19i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.52iT - 71T^{2} \) |
| 73 | \( 1 + (2.58 + 4.47i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.82 - 5.67i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6.49iT - 83T^{2} \) |
| 89 | \( 1 + (8.13 + 4.69i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 0.343T + 97T^{2} \) |
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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
−10.84928124148787209894710345473, −9.447741024504590585129494595396, −8.833927204240042821532032872282, −7.924066175458286055408880748099, −6.22063233677853371237672405841, −5.75549794325779548329343888726, −4.72510419779713927937185092980, −3.78577417644785496920324891297, −2.53198233215847057250333999643, −1.21949350257430154441633388067,
1.30533562626392632999025945760, 3.49600626301592587666105317424, 4.08055291898031623008189398427, 5.38241178715574874659457141249, 6.09927566396031200762258437098, 7.01098058283287138958754417620, 7.72810616611236378269147180730, 8.782692684445148756187135525901, 9.397367055306940833061774260679, 10.88767157025551551429795422185