L(s) = 1 | + (−1.35 − 0.388i)2-s + (−1.03 − 1.03i)3-s + (1.69 + 1.05i)4-s + (1.01 + 1.81i)6-s + 1.49i·7-s + (−1.89 − 2.09i)8-s − 0.836i·9-s + (0.423 − 0.423i)11-s + (−0.666 − 2.86i)12-s + (−1.85 − 1.85i)13-s + (0.581 − 2.03i)14-s + (1.76 + 3.58i)16-s + 6.50·17-s + (−0.325 + 1.13i)18-s + (−1.75 − 1.75i)19-s + ⋯ |
L(s) = 1 | + (−0.961 − 0.274i)2-s + (−0.600 − 0.600i)3-s + (0.849 + 0.528i)4-s + (0.412 + 0.742i)6-s + 0.565i·7-s + (−0.671 − 0.741i)8-s − 0.278i·9-s + (0.127 − 0.127i)11-s + (−0.192 − 0.827i)12-s + (−0.515 − 0.515i)13-s + (0.155 − 0.543i)14-s + (0.441 + 0.897i)16-s + 1.57·17-s + (−0.0766 + 0.268i)18-s + (−0.403 − 0.403i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.220290 - 0.486617i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.220290 - 0.486617i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.35 + 0.388i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (1.03 + 1.03i)T + 3iT^{2} \) |
| 7 | \( 1 - 1.49iT - 7T^{2} \) |
| 11 | \( 1 + (-0.423 + 0.423i)T - 11iT^{2} \) |
| 13 | \( 1 + (1.85 + 1.85i)T + 13iT^{2} \) |
| 17 | \( 1 - 6.50T + 17T^{2} \) |
| 19 | \( 1 + (1.75 + 1.75i)T + 19iT^{2} \) |
| 23 | \( 1 + 7.19iT - 23T^{2} \) |
| 29 | \( 1 + (6.57 + 6.57i)T + 29iT^{2} \) |
| 31 | \( 1 + 6.75T + 31T^{2} \) |
| 37 | \( 1 + (-1.95 + 1.95i)T - 37iT^{2} \) |
| 41 | \( 1 + 7.70iT - 41T^{2} \) |
| 43 | \( 1 + (6.13 - 6.13i)T - 43iT^{2} \) |
| 47 | \( 1 + 6.65T + 47T^{2} \) |
| 53 | \( 1 + (-5.29 + 5.29i)T - 53iT^{2} \) |
| 59 | \( 1 + (-5.91 + 5.91i)T - 59iT^{2} \) |
| 61 | \( 1 + (1.43 + 1.43i)T + 61iT^{2} \) |
| 67 | \( 1 + (6.35 + 6.35i)T + 67iT^{2} \) |
| 71 | \( 1 - 4.08iT - 71T^{2} \) |
| 73 | \( 1 - 2.43iT - 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + (-2.81 - 2.81i)T + 83iT^{2} \) |
| 89 | \( 1 - 10.5iT - 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−10.99152899868549800072789670863, −9.991627143872284768464419119790, −9.193259624391857060335555630469, −8.158498550486829818048651288946, −7.31061706207017113247593563061, −6.33638202172799762557368630997, −5.47418649436495744672437414036, −3.54285731455568619659733043898, −2.15142644921478320233778440715, −0.51385466495621445167485822159,
1.64493655352917586716770609483, 3.57252710036631325410919019182, 5.09135135122778766382599684831, 5.84828349863299825903127700479, 7.22248211643206765321531074470, 7.74070993370286496489511955473, 9.066877689945634379444288933440, 9.917772124492428224680384523344, 10.46023873568513272239096764698, 11.36744511597187403227125723963