L(s) = 1 | + (1.14 − 0.826i)2-s + 1.52i·3-s + (0.632 − 1.89i)4-s + (0.0967 − 2.23i)5-s + (1.25 + 1.74i)6-s + (−0.843 − 2.69i)8-s + 0.679·9-s + (−1.73 − 2.64i)10-s − 4.56i·11-s + (2.89 + 0.963i)12-s − 2.19·13-s + (3.40 + 0.147i)15-s + (−3.19 − 2.40i)16-s − 6.22·17-s + (0.779 − 0.561i)18-s + 3.83·19-s + ⋯ |
L(s) = 1 | + (0.811 − 0.584i)2-s + 0.879i·3-s + (0.316 − 0.948i)4-s + (0.0432 − 0.999i)5-s + (0.514 + 0.713i)6-s + (−0.298 − 0.954i)8-s + 0.226·9-s + (−0.549 − 0.835i)10-s − 1.37i·11-s + (0.834 + 0.278i)12-s − 0.608·13-s + (0.878 + 0.0380i)15-s + (−0.799 − 0.600i)16-s − 1.50·17-s + (0.183 − 0.132i)18-s + 0.879·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.47069 - 1.81329i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.47069 - 1.81329i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.14 + 0.826i)T \) |
| 5 | \( 1 + (-0.0967 + 2.23i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 1.52iT - 3T^{2} \) |
| 11 | \( 1 + 4.56iT - 11T^{2} \) |
| 13 | \( 1 + 2.19T + 13T^{2} \) |
| 17 | \( 1 + 6.22T + 17T^{2} \) |
| 19 | \( 1 - 3.83T + 19T^{2} \) |
| 23 | \( 1 - 0.430T + 23T^{2} \) |
| 29 | \( 1 + 0.473T + 29T^{2} \) |
| 31 | \( 1 - 7.59T + 31T^{2} \) |
| 37 | \( 1 + 8.44iT - 37T^{2} \) |
| 41 | \( 1 + 1.45iT - 41T^{2} \) |
| 43 | \( 1 - 8.58T + 43T^{2} \) |
| 47 | \( 1 + 4.48iT - 47T^{2} \) |
| 53 | \( 1 - 9.23iT - 53T^{2} \) |
| 59 | \( 1 + 3.13T + 59T^{2} \) |
| 61 | \( 1 + 5.71iT - 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 - 4.57iT - 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 6.20iT - 79T^{2} \) |
| 83 | \( 1 - 7.69iT - 83T^{2} \) |
| 89 | \( 1 - 9.32iT - 89T^{2} \) |
| 97 | \( 1 + 9.05T + 97T^{2} \) |
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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
−9.707014206296338958331955654332, −9.287328764027595688252145073942, −8.376659648928959368317311451859, −7.04913345054060719718509125250, −5.90258789872149738143769009450, −5.15304697045377402762981261006, −4.40063816118343567452818661534, −3.67219821074116647303507570455, −2.40527950664686895965153624787, −0.825828449723787310694362672260,
2.02134258650452765597460491954, 2.78980971191265311540899910876, 4.18884539763060867832554047917, 4.96372902290262285551229615499, 6.36071163432435031720456734047, 6.77370063842249558271217520304, 7.42540235601784689150584550247, 8.050881579850480061489194841235, 9.422083344440308807724647626924, 10.25044911313293856488680672593