L(s) = 1 | + (1.22 + 1.22i)7-s + (−1.22 + 1.22i)13-s − i·19-s + 31-s + (1.22 − 1.22i)43-s + 1.99i·49-s − 61-s + (−1.22 − 1.22i)67-s − 2i·79-s − 2.99·91-s + (−1.22 − 1.22i)97-s + i·109-s + ⋯ |
L(s) = 1 | + (1.22 + 1.22i)7-s + (−1.22 + 1.22i)13-s − i·19-s + 31-s + (1.22 − 1.22i)43-s + 1.99i·49-s − 61-s + (−1.22 − 1.22i)67-s − 2i·79-s − 2.99·91-s + (−1.22 − 1.22i)97-s + i·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.073134382\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.073134382\) |
\(L(1)\) |
|
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 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (1.22 - 1.22i)T - iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + iT - T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + 2iT - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
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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
−10.45650082203050754587648662324, −9.274047453835313919098768391461, −8.916301058786458780700295304279, −7.894100328866257288148004499645, −7.08475878432985113122910465677, −6.01040190620275579937618934991, −4.97757156259740452738938096395, −4.45645791053959658840702590127, −2.70906832935535461877131845257, −1.88259061974468479502304194211,
1.23196243106974681978028171987, 2.72978651449678672961196409348, 4.07139005917328997357403789229, 4.83848929404915519378501963603, 5.76283813340036200708337246177, 7.04773682711375936708027155857, 7.81775993782309514271072704526, 8.194432533002627210576222067735, 9.592621298271521156457216770300, 10.33800864541174707659782572476