L(s) = 1 | + (2.16 + 1.24i)2-s + (−1.41 + 2.44i)3-s + (2.11 + 3.66i)4-s + (−6.10 + 3.52i)6-s + (1.64 − 0.952i)7-s + 5.55i·8-s + (−2.49 − 4.32i)9-s + (0.926 + 0.534i)11-s − 11.9·12-s + (−1.40 − 3.32i)13-s + 4.75·14-s + (−2.70 + 4.69i)16-s + (−0.318 − 0.551i)17-s − 12.4i·18-s + (4.96 − 2.86i)19-s + ⋯ |
L(s) = 1 | + (1.52 + 0.882i)2-s + (−0.816 + 1.41i)3-s + (1.05 + 1.83i)4-s + (−2.49 + 1.43i)6-s + (0.623 − 0.360i)7-s + 1.96i·8-s + (−0.831 − 1.44i)9-s + (0.279 + 0.161i)11-s − 3.44·12-s + (−0.388 − 0.921i)13-s + 1.27·14-s + (−0.677 + 1.17i)16-s + (−0.0772 − 0.133i)17-s − 2.93i·18-s + (1.13 − 0.657i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.747953 + 2.27459i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.747953 + 2.27459i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 + (1.40 + 3.32i)T \) |
good | 2 | \( 1 + (-2.16 - 1.24i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (1.41 - 2.44i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (-1.64 + 0.952i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.926 - 0.534i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (0.318 + 0.551i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.96 + 2.86i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.90 - 3.30i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.72 - 8.18i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 1.46iT - 31T^{2} \) |
| 37 | \( 1 + (0.655 + 0.378i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.232 + 0.133i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.318 + 0.551i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 9.44iT - 47T^{2} \) |
| 53 | \( 1 - 6.99T + 53T^{2} \) |
| 59 | \( 1 + (0.641 - 0.370i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.09 + 3.63i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.01 - 4.04i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.45 + 4.88i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 3.71iT - 73T^{2} \) |
| 79 | \( 1 + 9.31T + 79T^{2} \) |
| 83 | \( 1 + 5.11iT - 83T^{2} \) |
| 89 | \( 1 + (10.8 + 6.28i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.65 + 2.11i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.92191765621933106523598788727, −11.32040997031961078290408665436, −10.36907500918585910812144394030, −9.281938790362022923947035859116, −7.78719034558749697235372500759, −6.86123774870959434646419986393, −5.42412229717834649910326435764, −5.24854098412072861084782126689, −4.17598626085751582157722544404, −3.28194816282593312327091037268,
1.43987374147672883783489464617, 2.41214273734519342401260817594, 4.12209896379259501707295438139, 5.30896041762375586769244589500, 6.02303007206520616175365430016, 6.93784967045033372616283712288, 8.086027934887792280552780815090, 9.765259141305849385805007211736, 11.10041429560829759631707675707, 11.60956957718450023644200388015