L(s) = 1 | + (1.22 + 0.710i)2-s + (−1.09 + 1.09i)3-s + (0.991 + 1.73i)4-s + (−2.11 + 0.560i)6-s − 0.973i·7-s + (−0.0202 + 2.82i)8-s + 0.616i·9-s + (1.40 + 1.40i)11-s + (−2.97 − 0.813i)12-s + (−4.60 + 4.60i)13-s + (0.691 − 1.19i)14-s + (−2.03 + 3.44i)16-s + 0.490·17-s + (−0.438 + 0.754i)18-s + (4.54 − 4.54i)19-s + ⋯ |
L(s) = 1 | + (0.864 + 0.502i)2-s + (−0.630 + 0.630i)3-s + (0.495 + 0.868i)4-s + (−0.861 + 0.228i)6-s − 0.368i·7-s + (−0.00714 + 0.999i)8-s + 0.205i·9-s + (0.424 + 0.424i)11-s + (−0.859 − 0.234i)12-s + (−1.27 + 1.27i)13-s + (0.184 − 0.318i)14-s + (−0.508 + 0.861i)16-s + 0.118·17-s + (−0.103 + 0.177i)18-s + (1.04 − 1.04i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.601 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.601 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.763961 + 1.53074i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.763961 + 1.53074i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 - 0.710i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (1.09 - 1.09i)T - 3iT^{2} \) |
| 7 | \( 1 + 0.973iT - 7T^{2} \) |
| 11 | \( 1 + (-1.40 - 1.40i)T + 11iT^{2} \) |
| 13 | \( 1 + (4.60 - 4.60i)T - 13iT^{2} \) |
| 17 | \( 1 - 0.490T + 17T^{2} \) |
| 19 | \( 1 + (-4.54 + 4.54i)T - 19iT^{2} \) |
| 23 | \( 1 + 1.94iT - 23T^{2} \) |
| 29 | \( 1 + (3.74 - 3.74i)T - 29iT^{2} \) |
| 31 | \( 1 - 4.29T + 31T^{2} \) |
| 37 | \( 1 + (-4.55 - 4.55i)T + 37iT^{2} \) |
| 41 | \( 1 + 10.1iT - 41T^{2} \) |
| 43 | \( 1 + (1.79 + 1.79i)T + 43iT^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + (-5.61 - 5.61i)T + 53iT^{2} \) |
| 59 | \( 1 + (-8.44 - 8.44i)T + 59iT^{2} \) |
| 61 | \( 1 + (-3.01 + 3.01i)T - 61iT^{2} \) |
| 67 | \( 1 + (-7.07 + 7.07i)T - 67iT^{2} \) |
| 71 | \( 1 - 0.897iT - 71T^{2} \) |
| 73 | \( 1 + 9.71iT - 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + (-0.815 + 0.815i)T - 83iT^{2} \) |
| 89 | \( 1 - 1.12iT - 89T^{2} \) |
| 97 | \( 1 + 7.54T + 97T^{2} \) |
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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
−11.75936849627635419015954732096, −10.90410007816195286903336590316, −9.870581390003754676015240772227, −8.891140707107413111254154842736, −7.37995213595495345207476814261, −6.95086354382372653240471766504, −5.59602036450077326790263458304, −4.74770830022145913039485483591, −4.08154508448976583609043571179, −2.42707570903185804227806504943,
0.961020376227969477885819242782, 2.63127935758387584787797836777, 3.84428435021896002203205559320, 5.41381559938594618642950557527, 5.79721368633680145175879683435, 6.95899991149356269361487842691, 7.897383990059603863345719540102, 9.526752671093889369624951775048, 10.14979765692183972712375253519, 11.45016354673827687989063735020