L(s) = 1 | + (−1 + i)13-s + (1.41 − 1.41i)17-s + 1.41·29-s + (1 + i)37-s − 1.41i·41-s + i·49-s + (1.41 + 1.41i)53-s + (1 − i)73-s − 1.41·89-s + (1 + i)97-s − 1.41i·101-s + ⋯ |
L(s) = 1 | + (−1 + i)13-s + (1.41 − 1.41i)17-s + 1.41·29-s + (1 + i)37-s − 1.41i·41-s + i·49-s + (1.41 + 1.41i)53-s + (1 − i)73-s − 1.41·89-s + (1 + i)97-s − 1.41i·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.289309924\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.289309924\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (1 - i)T - iT^{2} \) |
| 17 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - 1.41T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-1 - i)T + iT^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1 + i)T - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + 1.41T + T^{2} \) |
| 97 | \( 1 + (-1 - i)T + iT^{2} \) |
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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
−8.834183045858896215590023219831, −7.85867063911802813772846974296, −7.30081693918475434795623116413, −6.63065872678524702680547266745, −5.67988046107564714656975990578, −4.90449726426965376192210600265, −4.25824456212995958992485879422, −3.08710023255538321158036724516, −2.39260388633025422999209861023, −1.03496738874621367978271446123,
1.00610473011614275017122542910, 2.31431995466167475123812049611, 3.20289258474364028574680896810, 4.04770330587375452454865044459, 5.05493350352688407288564698729, 5.66979806296969947094178306581, 6.45791472496820676369570347129, 7.35541345586205908549329858971, 8.077832017830271848652094282791, 8.463188702016049137277626859858