L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s − 1.00i·4-s + 1.00·6-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (−0.707 + 0.707i)12-s − 1.00·16-s + (1.41 − 1.41i)17-s + (−0.707 − 0.707i)18-s − 2i·19-s − 1.00i·24-s + (0.707 − 0.707i)27-s + (0.707 − 0.707i)32-s + 2.00i·34-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s − 1.00i·4-s + 1.00·6-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (−0.707 + 0.707i)12-s − 1.00·16-s + (1.41 − 1.41i)17-s + (−0.707 − 0.707i)18-s − 2i·19-s − 1.00i·24-s + (0.707 − 0.707i)27-s + (0.707 − 0.707i)32-s + 2.00i·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5135013168\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5135013168\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 19 | \( 1 + 2iT - T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−10.83807134104021538919445611166, −9.805113774244565345464745877110, −9.028749079948875148683035615096, −7.891029457169636612642187342351, −7.24158237298587336364111639163, −6.52609314592678570021388224505, −5.44616737155818135769791162795, −4.80159844471133660201479248304, −2.60587304832788196123186125509, −0.934443192434912603597736463747,
1.52492142328152317322830966531, 3.39863170052532323827461680497, 4.04886870152949275112666470514, 5.45707401313710585920192779646, 6.37184887812364498097616006368, 7.71542836175261784312940339271, 8.445017224444576913761884596745, 9.542691856045689371242719581915, 10.26284025884374299403228636652, 10.62300218863255284045641217800