L(s) = 1 | + 1.41·3-s + (0.707 + 0.707i)5-s + 1.00·9-s + (−0.707 − 0.707i)13-s + (1.00 + 1.00i)15-s + 1.41i·17-s + (0.707 + 0.707i)19-s + i·23-s − 29-s + (−0.707 − 0.707i)31-s + (1 − i)37-s + (−1.00 − 1.00i)39-s − i·43-s + (0.707 + 0.707i)45-s + (0.707 − 0.707i)47-s + ⋯ |
L(s) = 1 | + 1.41·3-s + (0.707 + 0.707i)5-s + 1.00·9-s + (−0.707 − 0.707i)13-s + (1.00 + 1.00i)15-s + 1.41i·17-s + (0.707 + 0.707i)19-s + i·23-s − 29-s + (−0.707 − 0.707i)31-s + (1 − i)37-s + (−1.00 − 1.00i)39-s − i·43-s + (0.707 + 0.707i)45-s + (0.707 − 0.707i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.117169872\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.117169872\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 - 1.41T + T^{2} \) |
| 5 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 17 | \( 1 - 1.41iT - T^{2} \) |
| 19 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 23 | \( 1 - iT - T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + iT - T^{2} \) |
| 47 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 + (1 + i)T + iT^{2} \) |
| 71 | \( 1 + (1 + i)T + iT^{2} \) |
| 73 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 89 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 97 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
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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
−9.259595317374064530416293645582, −8.293709979583084834011616115125, −7.71436284554181993969117046501, −7.13260258721512848625763794384, −5.96000962558609064104567865223, −5.45227604154539325389210406615, −3.92624054978879261851738218880, −3.43487267265167992252872095127, −2.41530868358777968719648294845, −1.81678024406073891736278018308,
1.37124271307593974350886651711, 2.47852947228702082011571428318, 2.99989622638371912573610358766, 4.28816444566749275919479489651, 4.93849624922076652841012025673, 5.84463365836190152423035512698, 7.11203608918565685806731698020, 7.43292022374580725509501548629, 8.540076325582319403604566526151, 9.036449901136071275297841152563