| L(s) = 1 | + i·2-s + (1.06 + 1.36i)3-s − 4-s + (−1.36 + 1.06i)6-s + (−2.62 − 0.294i)7-s − i·8-s + (−0.716 + 2.91i)9-s + 1.43i·11-s + (−1.06 − 1.36i)12-s + 4.73i·13-s + (0.294 − 2.62i)14-s + 16-s − 2.59·17-s + (−2.91 − 0.716i)18-s − 2.72i·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (0.616 + 0.786i)3-s − 0.5·4-s + (−0.556 + 0.436i)6-s + (−0.993 − 0.111i)7-s − 0.353i·8-s + (−0.238 + 0.971i)9-s + 0.431i·11-s + (−0.308 − 0.393i)12-s + 1.31i·13-s + (0.0786 − 0.702i)14-s + 0.250·16-s − 0.630·17-s + (−0.686 − 0.168i)18-s − 0.625i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9258282239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9258282239\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-1.06 - 1.36i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.62 + 0.294i)T \) |
| good | 11 | \( 1 - 1.43iT - 11T^{2} \) |
| 13 | \( 1 - 4.73iT - 13T^{2} \) |
| 17 | \( 1 + 2.59T + 17T^{2} \) |
| 19 | \( 1 + 2.72iT - 19T^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 5.91iT - 31T^{2} \) |
| 37 | \( 1 + 2.39T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 - 0.432T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 5.69iT - 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 + 1.05iT - 61T^{2} \) |
| 67 | \( 1 - 9.82T + 67T^{2} \) |
| 71 | \( 1 - 13.2iT - 71T^{2} \) |
| 73 | \( 1 - 7.58iT - 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 3.90T + 89T^{2} \) |
| 97 | \( 1 - 14.0iT - 97T^{2} \) |
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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
−9.980422909853186646555510146037, −9.464786719187922884062096932682, −8.919044256014206038000004622368, −7.950135399047520185397953644238, −6.98498608676597128675467454151, −6.34906208496827973546298057636, −5.11479941338378574445163532276, −4.28871606804464114954447990938, −3.47985462342393769675721264127, −2.19051490453833884857933134151,
0.36552122907014934422579723481, 1.84395676891438595083023054265, 3.13818200887365177764750535851, 3.43568291666349052858495920065, 5.04423244209913270645696204113, 6.15930719594166711386602627217, 6.86248353468359370330285762869, 8.033057446502522348988653427425, 8.585068329098052110711348942610, 9.430833267376533388808976988367
