L(s) = 1 | + i·3-s + (−1 + 2i)5-s + i·7-s − 9-s − 2·11-s − 2i·13-s + (−2 − i)15-s − 6·19-s − 21-s + (−3 − 4i)25-s − i·27-s + 6·29-s − 10·31-s − 2i·33-s + (−2 − i)35-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (−0.447 + 0.894i)5-s + 0.377i·7-s − 0.333·9-s − 0.603·11-s − 0.554i·13-s + (−0.516 − 0.258i)15-s − 1.37·19-s − 0.218·21-s + (−0.600 − 0.800i)25-s − 0.192i·27-s + 1.11·29-s − 1.79·31-s − 0.348i·33-s + (−0.338 − 0.169i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (1 - 2i)T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−9.075902734013321037121669723567, −8.293022342854331376389214727987, −7.60908435745312751754224126055, −6.63604034244272924016501614831, −5.86716485607372703701422018418, −4.92804077960053628742895460071, −3.96745148901236981397136469975, −3.08496735893623022512876683261, −2.21634261915250169850858491996, 0,
1.36160676921459231064328513543, 2.50418009415214929313552227416, 3.87140850746027277809692761816, 4.59349969555618474721873585470, 5.54807028692432784657098872226, 6.45718057582738386171064435169, 7.36958787083789198539248580781, 7.973883388871348442660281323408, 8.777198821341343106205046082890