L(s) = 1 | − i·2-s − i·3-s − 4-s + (−0.539 + 2.17i)5-s − 6-s − 1.07i·7-s + i·8-s − 9-s + (2.17 + 0.539i)10-s + 6.34·11-s + i·12-s − 3.41i·13-s − 1.07·14-s + (2.17 + 0.539i)15-s + 16-s − 5.41i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.241 + 0.970i)5-s − 0.408·6-s − 0.407i·7-s + 0.353i·8-s − 0.333·9-s + (0.686 + 0.170i)10-s + 1.91·11-s + 0.288i·12-s − 0.948i·13-s − 0.288·14-s + (0.560 + 0.139i)15-s + 0.250·16-s − 1.31i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.833140 - 1.06547i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.833140 - 1.06547i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (0.539 - 2.17i)T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 1.07iT - 7T^{2} \) |
| 11 | \( 1 - 6.34T + 11T^{2} \) |
| 13 | \( 1 + 3.41iT - 13T^{2} \) |
| 17 | \( 1 + 5.41iT - 17T^{2} \) |
| 23 | \( 1 + 6.34iT - 23T^{2} \) |
| 29 | \( 1 - 0.340T + 29T^{2} \) |
| 31 | \( 1 - 1.07T + 31T^{2} \) |
| 37 | \( 1 - 3.41iT - 37T^{2} \) |
| 41 | \( 1 - 7.60T + 41T^{2} \) |
| 43 | \( 1 - 11.1iT - 43T^{2} \) |
| 47 | \( 1 + 6.34iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 0.738T + 59T^{2} \) |
| 61 | \( 1 + 2.68T + 61T^{2} \) |
| 67 | \( 1 - 2.83iT - 67T^{2} \) |
| 71 | \( 1 + 2.83T + 71T^{2} \) |
| 73 | \( 1 + 6.83iT - 73T^{2} \) |
| 79 | \( 1 - 1.07T + 79T^{2} \) |
| 83 | \( 1 + 0.894iT - 83T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 97 | \( 1 + 3.65iT - 97T^{2} \) |
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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
−10.62029088868600397900563173691, −9.767912359554629135645061041020, −8.822681838426324220010912001987, −7.76205384826580137483446425049, −6.85858959030907383185731110403, −6.14232154303780458316053637612, −4.56421428813082819501461395220, −3.49289241883032846454710924905, −2.50701061230536849326127293194, −0.883583711262000729536427883987,
1.53183109142560054556249760847, 3.88730176969095619246564633366, 4.25119854084832143991695701691, 5.56564875337071980002908405314, 6.29470379124610929477660213149, 7.44340415357525795642276132126, 8.651568588817250227679040094782, 9.044600509040484211592580696179, 9.692532806885252921889493876208, 11.06204503847437797458908113920