L(s) = 1 | + (−0.707 − 0.707i)3-s + (−0.811 + 2.08i)5-s + (3.03 + 3.03i)7-s + 1.00i·9-s + 0.215i·11-s + (−0.104 − 0.104i)13-s + (2.04 − 0.899i)15-s + (0.838 + 0.838i)17-s + 3.62·19-s − 4.28i·21-s + (−4.41 + 1.87i)23-s + (−3.68 − 3.38i)25-s + (0.707 − 0.707i)27-s + 5.72i·29-s − 0.698·31-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (−0.362 + 0.931i)5-s + (1.14 + 1.14i)7-s + 0.333i·9-s + 0.0648i·11-s + (−0.0288 − 0.0288i)13-s + (0.528 − 0.232i)15-s + (0.203 + 0.203i)17-s + 0.832·19-s − 0.935i·21-s + (−0.920 + 0.390i)23-s + (−0.736 − 0.676i)25-s + (0.136 − 0.136i)27-s + 1.06i·29-s − 0.125·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.248471626\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.248471626\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.811 - 2.08i)T \) |
| 23 | \( 1 + (4.41 - 1.87i)T \) |
good | 7 | \( 1 + (-3.03 - 3.03i)T + 7iT^{2} \) |
| 11 | \( 1 - 0.215iT - 11T^{2} \) |
| 13 | \( 1 + (0.104 + 0.104i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.838 - 0.838i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.62T + 19T^{2} \) |
| 29 | \( 1 - 5.72iT - 29T^{2} \) |
| 31 | \( 1 + 0.698T + 31T^{2} \) |
| 37 | \( 1 + (3.51 + 3.51i)T + 37iT^{2} \) |
| 41 | \( 1 + 1.54T + 41T^{2} \) |
| 43 | \( 1 + (1.20 - 1.20i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.22 - 3.22i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.708 + 0.708i)T - 53iT^{2} \) |
| 59 | \( 1 - 2.79iT - 59T^{2} \) |
| 61 | \( 1 + 5.64iT - 61T^{2} \) |
| 67 | \( 1 + (-5.33 - 5.33i)T + 67iT^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + (-9.19 - 9.19i)T + 73iT^{2} \) |
| 79 | \( 1 - 0.634T + 79T^{2} \) |
| 83 | \( 1 + (3.22 - 3.22i)T - 83iT^{2} \) |
| 89 | \( 1 - 9.91T + 89T^{2} \) |
| 97 | \( 1 + (-11.7 - 11.7i)T + 97iT^{2} \) |
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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
−9.911973139693294071733944461328, −8.877312933119482569134039904966, −8.032049367285919594164196037285, −7.48090694880992685439521120707, −6.54139935358099066533167699502, −5.64358919461574447809597636263, −5.01047472239063645939130396523, −3.71510203006079601163300313613, −2.57486380141497029458200898815, −1.59542791012473412491525898755,
0.55842634689219894799505839044, 1.69327133521020758354781287955, 3.54217522946710877613573680339, 4.39296391128311487820656513746, 4.93071335855645545219677402584, 5.81704433887367348217043389917, 7.04777836826621236980006248491, 7.86402495885285630022996606038, 8.380256742995131580555222859508, 9.440875929334299827070162003814