L(s) = 1 | − 2-s + (1.03 + 1.38i)3-s + 4-s + (−1.98 − 1.02i)5-s + (−1.03 − 1.38i)6-s − 8-s + (−0.844 + 2.87i)9-s + (1.98 + 1.02i)10-s + 1.27i·11-s + (1.03 + 1.38i)12-s − 1.32·13-s + (−0.643 − 3.81i)15-s + 16-s + 5.76i·17-s + (0.844 − 2.87i)18-s − 2.44i·19-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.599 + 0.800i)3-s + 0.5·4-s + (−0.888 − 0.457i)5-s + (−0.423 − 0.566i)6-s − 0.353·8-s + (−0.281 + 0.959i)9-s + (0.628 + 0.323i)10-s + 0.385i·11-s + (0.299 + 0.400i)12-s − 0.366·13-s + (−0.166 − 0.986i)15-s + 0.250·16-s + 1.39i·17-s + (0.199 − 0.678i)18-s − 0.560i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2420611066\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2420611066\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-1.03 - 1.38i)T \) |
| 5 | \( 1 + (1.98 + 1.02i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 1.27iT - 11T^{2} \) |
| 13 | \( 1 + 1.32T + 13T^{2} \) |
| 17 | \( 1 - 5.76iT - 17T^{2} \) |
| 19 | \( 1 + 2.44iT - 19T^{2} \) |
| 23 | \( 1 + 0.680T + 23T^{2} \) |
| 29 | \( 1 + 6.12iT - 29T^{2} \) |
| 31 | \( 1 + 6.94iT - 31T^{2} \) |
| 37 | \( 1 - 4.67iT - 37T^{2} \) |
| 41 | \( 1 + 12.3T + 41T^{2} \) |
| 43 | \( 1 - 1.87iT - 43T^{2} \) |
| 47 | \( 1 + 7.97iT - 47T^{2} \) |
| 53 | \( 1 + 6.98T + 53T^{2} \) |
| 59 | \( 1 + 9.65T + 59T^{2} \) |
| 61 | \( 1 - 4.22iT - 61T^{2} \) |
| 67 | \( 1 - 12.9iT - 67T^{2} \) |
| 71 | \( 1 + 5.09iT - 71T^{2} \) |
| 73 | \( 1 + 16.4T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 - 3.33iT - 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 2.90T + 97T^{2} \) |
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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
−9.920724138719497592373118497605, −9.071860745273965678959757655147, −8.357768234676746995852333591028, −7.906926947626572298419816679717, −7.02660214430139551767710467006, −5.82862383698027603017945376960, −4.68962096143034793127732331624, −4.02328221938318401855127649062, −3.00377939831774455718637821597, −1.77886442252259381080624039906,
0.11018535004683050139312274470, 1.54003183083965174872279987716, 2.89637187934301115037096658060, 3.40251204218479909356310669867, 4.85016938604959613516060858493, 6.14661033492671735814956143136, 7.06798617489908826780641218957, 7.39906175699801336992764887212, 8.268841333419574939763548452580, 8.860897033733855592632476085725