L(s) = 1 | + (0.366 + 1.36i)2-s + 1.73·3-s + (−1.73 + i)4-s − i·5-s + (0.633 + 2.36i)6-s + (1.73 + 2i)7-s + (−2 − 1.99i)8-s + (1.36 − 0.366i)10-s + 0.267i·11-s + (−2.99 + 1.73i)12-s − 0.464i·13-s + (−2.09 + 3.09i)14-s − 1.73i·15-s + (1.99 − 3.46i)16-s − 6.46i·17-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)2-s + 1.00·3-s + (−0.866 + 0.5i)4-s − 0.447i·5-s + (0.258 + 0.965i)6-s + (0.654 + 0.755i)7-s + (−0.707 − 0.707i)8-s + (0.431 − 0.115i)10-s + 0.0807i·11-s + (−0.866 + 0.499i)12-s − 0.128i·13-s + (−0.560 + 0.827i)14-s − 0.447i·15-s + (0.499 − 0.866i)16-s − 1.56i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.21759 + 0.866793i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.21759 + 0.866793i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.366 - 1.36i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (-1.73 - 2i)T \) |
good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 11 | \( 1 - 0.267iT - 11T^{2} \) |
| 13 | \( 1 + 0.464iT - 13T^{2} \) |
| 17 | \( 1 + 6.46iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 1.46iT - 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 9.46T + 37T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + 1.73T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 - 9.46iT - 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 7.46iT - 71T^{2} \) |
| 73 | \( 1 - 12.9iT - 73T^{2} \) |
| 79 | \( 1 - 14.6iT - 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - 2.53iT - 89T^{2} \) |
| 97 | \( 1 + 13.3iT - 97T^{2} \) |
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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
−13.78306444111445986732990603765, −12.60580569319714463480723518746, −11.65562543394615297157962779327, −9.801244868834044834706517277330, −8.613269021347616704676850419010, −8.414513807103850625966744137325, −7.04673002360075964241314787950, −5.56255109926202987549407722580, −4.46685483100487349295772919683, −2.74950593616105688667329607210,
2.00317868926136743976938576562, 3.43956459793713285693466001337, 4.53073752494076263023383963801, 6.35384541808083002847177531649, 8.135029386060292695619016365753, 8.693167579331408433662177314717, 10.22206421234496715825580858842, 10.73768373547095498425043076828, 11.94128125198125226072746342311, 13.10387897814592251589788565736