L(s) = 1 | + i·2-s + (0.707 − 0.707i)3-s − 4-s + (−0.707 + 0.707i)5-s + (0.707 + 0.707i)6-s − i·8-s − 1.00i·9-s + (−0.707 − 0.707i)10-s + (−0.707 + 0.707i)12-s + 1.41·13-s + 1.00i·15-s + 16-s − i·17-s + 1.00·18-s + (0.707 − 0.707i)20-s + ⋯ |
L(s) = 1 | + i·2-s + (0.707 − 0.707i)3-s − 4-s + (−0.707 + 0.707i)5-s + (0.707 + 0.707i)6-s − i·8-s − 1.00i·9-s + (−0.707 − 0.707i)10-s + (−0.707 + 0.707i)12-s + 1.41·13-s + 1.00i·15-s + 16-s − i·17-s + 1.00·18-s + (0.707 − 0.707i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.248468891\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.248468891\) |
\(L(1)\) |
|
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 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 17 | \( 1 + iT \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 - 1.41T + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-1 - i)T + iT^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 + (-1 + i)T - iT^{2} \) |
| 37 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 41 | \( 1 - iT^{2} \) |
| 43 | \( 1 - 1.41iT - T^{2} \) |
| 47 | \( 1 - 2T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 1.41iT - T^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 - 1.41T + T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + (1 + i)T + iT^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−9.133397702211611600355797478456, −8.331769540936104923204875427142, −7.84681764172367147729507190556, −7.05380884070122702444287709903, −6.56857189516605254308522015996, −5.72892494726628488942530674530, −4.49680513097439473314905754589, −3.56791751927935149203697644466, −2.89965848028988088040174761889, −1.10731132304985474754128575653,
1.21884650021462173129695201942, 2.47331469007647382926837469520, 3.70876248132783035977367448230, 3.89641536094551759624369467164, 4.91244654380516258994763354782, 5.66762581524266877450205463682, 7.13590116390657978226553630620, 8.258620167047191786807995443237, 8.780944325209169975958455086589, 8.900384388675439326490535721591