L(s) = 1 | + (0.258 − 0.965i)2-s + (0.707 + 0.707i)3-s + (−0.866 − 0.499i)4-s + (−1.22 + 1.22i)5-s + (0.866 − 0.500i)6-s − i·7-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (0.866 + 1.49i)10-s + (0.707 − 0.707i)11-s + (−0.258 − 0.965i)12-s + (−1.36 − 1.36i)13-s + (−0.965 − 0.258i)14-s − 1.73·15-s + (0.500 + 0.866i)16-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)2-s + (0.707 + 0.707i)3-s + (−0.866 − 0.499i)4-s + (−1.22 + 1.22i)5-s + (0.866 − 0.500i)6-s − i·7-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (0.866 + 1.49i)10-s + (0.707 − 0.707i)11-s + (−0.258 − 0.965i)12-s + (−1.36 − 1.36i)13-s + (−0.965 − 0.258i)14-s − 1.73·15-s + (0.500 + 0.866i)16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8587494621\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8587494621\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (-0.707 + 0.707i)T \) |
good | 5 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 13 | \( 1 + (1.36 + 1.36i)T + iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - 0.517T + T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 61 | \( 1 + (1 + i)T + iT^{2} \) |
| 67 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.426853273959809630723666746666, −7.74751511545583538733066476775, −7.31095367564695003574834085278, −6.14070393408769990750303578291, −5.02775923342403875756170047086, −4.15861064909417696152723904957, −3.72322814432332371487620243615, −3.04241612906487335797525552313, −2.36517567939335471774692809089, −0.42417554569472267962379508679,
1.44484280190024430179114480180, 2.65640222590014169528380262948, 3.89652396992189535018648590980, 4.40995177098175511337591324714, 5.10252964018307951363042408596, 6.15196923754458824164016920585, 6.97820647274075206689326828975, 7.49481294760058108783846861132, 8.150858420687594597322083057239, 8.854002750797542736254200404742