L(s) = 1 | + (0.866 − 0.5i)4-s + (0.133 − 0.5i)7-s + i·13-s + (0.499 − 0.866i)16-s + (0.5 + 0.133i)19-s + (−0.133 − 0.5i)28-s + (1 − i)31-s + (1.36 − 0.366i)37-s + (−1.5 + 0.866i)43-s + (0.633 + 0.366i)49-s + (0.5 + 0.866i)52-s − 0.999i·64-s + (−0.5 − 1.86i)67-s + (−1.36 − 1.36i)73-s + (0.5 − 0.133i)76-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)4-s + (0.133 − 0.5i)7-s + i·13-s + (0.499 − 0.866i)16-s + (0.5 + 0.133i)19-s + (−0.133 − 0.5i)28-s + (1 − i)31-s + (1.36 − 0.366i)37-s + (−1.5 + 0.866i)43-s + (0.633 + 0.366i)49-s + (0.5 + 0.866i)52-s − 0.999i·64-s + (−0.5 − 1.86i)67-s + (−1.36 − 1.36i)73-s + (0.5 − 0.133i)76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.594388826\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.594388826\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
good | 2 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 7 | \( 1 + (-0.133 + 0.5i)T + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.133i)T + (0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-1 + i)T - iT^{2} \) |
| 37 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 43 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 73 | \( 1 + (1.36 + 1.36i)T + iT^{2} \) |
| 79 | \( 1 + 1.73T + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
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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
−8.986147791982580246177516066623, −7.84926143049258814668763774540, −7.43200606746497452884646501943, −6.46797066758494080413790335414, −6.08404192862290561879859424761, −4.97755577024953136950116852946, −4.22774791812799764003122329312, −3.12298374762945814591092715890, −2.14965071881703208555960811316, −1.15776493742152623949957724481,
1.38787688414008176262026586480, 2.63722059660031503358850603036, 3.12270060869036489220381493282, 4.23477761001788720688829132520, 5.32338199147974026252699449427, 5.94745639268109948186034442936, 6.85427667852945022254212330624, 7.45575901201244678432530610215, 8.333937610056092271649498853496, 8.685253513737055619685529274108