L(s) = 1 | + (−0.707 + 1.22i)3-s + (0.5 + 0.866i)4-s − i·5-s + (−0.499 − 0.866i)9-s − 1.41·12-s + (0.258 − 0.965i)13-s + (1.22 + 0.707i)15-s + (−0.499 + 0.866i)16-s + (0.965 + 1.67i)17-s + (0.866 − 0.5i)19-s + (0.866 − 0.5i)20-s + (−0.965 + 1.67i)23-s − 25-s + i·31-s + (0.5 − 0.866i)36-s + (0.448 + 0.258i)37-s + ⋯ |
L(s) = 1 | + (−0.707 + 1.22i)3-s + (0.5 + 0.866i)4-s − i·5-s + (−0.499 − 0.866i)9-s − 1.41·12-s + (0.258 − 0.965i)13-s + (1.22 + 0.707i)15-s + (−0.499 + 0.866i)16-s + (0.965 + 1.67i)17-s + (0.866 − 0.5i)19-s + (0.866 − 0.5i)20-s + (−0.965 + 1.67i)23-s − 25-s + i·31-s + (0.5 − 0.866i)36-s + (0.448 + 0.258i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.007065157\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.007065157\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + iT \) |
| 13 | \( 1 + (-0.258 + 0.965i)T \) |
| 31 | \( 1 - iT \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - 0.517T + T^{2} \) |
| 59 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + 1.41iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.93iT - T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
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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
−9.646140613966947901773138652278, −8.832229912182336785920757455156, −7.995874404226894210228570049272, −7.50487463970051200309949329040, −5.94979274334964810546871126064, −5.66829992603975291566297816575, −4.66305794505722932066907730971, −3.82926726313611988973756129841, −3.23245836841951289615672778490, −1.51002879591096897703847883456,
0.843095302991820275150771690498, 2.04318581810753957681434842079, 2.80206683807592323181802230002, 4.29373826296101661210810823302, 5.58523936419869861922838602817, 6.04336193449732040780171928584, 6.74305134214445691192220350708, 7.31618213812144548618838480913, 7.87678972969927382532849402871, 9.403248828606640074364497237015