L(s) = 1 | + 1.41·2-s + 1.00·4-s + (0.707 + 0.707i)5-s + 7-s + (1.00 + 1.00i)10-s + (−0.707 − 0.707i)11-s + 13-s + 1.41·14-s − 0.999·16-s + (−0.707 − 0.707i)17-s + (−1 + i)19-s + (0.707 + 0.707i)20-s + (−1.00 − 1.00i)22-s + 1.00i·25-s + 1.41·26-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1.00·4-s + (0.707 + 0.707i)5-s + 7-s + (1.00 + 1.00i)10-s + (−0.707 − 0.707i)11-s + 13-s + 1.41·14-s − 0.999·16-s + (−0.707 − 0.707i)17-s + (−1 + i)19-s + (0.707 + 0.707i)20-s + (−1.00 − 1.00i)22-s + 1.00i·25-s + 1.41·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.621069612\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.621069612\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 17 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 19 | \( 1 + (1 - i)T - iT^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (1 + i)T + iT^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 43 | \( 1 + (-1 - i)T + iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 59 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 + iT - T^{2} \) |
| 71 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 - iT - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 97 | \( 1 + iT - T^{2} \) |
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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
−9.548872941764306555137014615841, −8.671300218392043534386980824662, −7.83707226537522677770431605005, −6.82216087102697696642171591512, −5.91510672290869245240609005779, −5.62976608231496628985333799127, −4.56628536915854981049288268779, −3.76506678925825644526394814190, −2.75974532793262421944558064514, −1.90240408876818796683573565821,
1.71386607348962492336321127695, 2.52429034366525709588862962788, 3.92316846370931628506804467984, 4.66301570028912847544003114390, 5.14626690480872888404158313723, 6.01391869353708977751589637823, 6.70071644723532245545908842217, 7.86477467478830299602985925043, 8.764118055006403787122376107134, 9.227729764622209036802139840665