L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.292 + 0.707i)11-s − 1.00·14-s − 1.00·16-s + (−0.292 − 0.707i)22-s + (−1 + i)23-s + (0.707 + 0.707i)25-s + (0.707 − 0.707i)28-s + (0.292 + 0.707i)29-s + (0.707 − 0.707i)32-s + (−0.707 − 0.292i)37-s + (0.292 − 0.707i)43-s + (0.707 + 0.292i)44-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (−0.292 + 0.707i)11-s − 1.00·14-s − 1.00·16-s + (−0.292 − 0.707i)22-s + (−1 + i)23-s + (0.707 + 0.707i)25-s + (0.707 − 0.707i)28-s + (0.292 + 0.707i)29-s + (0.707 − 0.707i)32-s + (−0.707 − 0.292i)37-s + (0.292 − 0.707i)43-s + (0.707 + 0.292i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8027820168\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8027820168\) |
\(L(1)\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 5 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + (1 - i)T - iT^{2} \) |
| 29 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 67 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 71 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.428855145517124773901186913696, −8.670582822363449051221061245407, −8.068421173697305662831900416126, −7.28309416669652983408774907885, −6.61777728932487910636892613954, −5.34502549523002123212430684397, −5.27605706992106107318859452454, −3.96738977559825674273755082752, −2.41404507905652270447695420645, −1.50608241021487154035623972960,
0.77558862603944845971577500018, 2.05677794648914722140594652200, 3.06199598888594220859908858667, 4.11347635101029494158171089117, 4.77317248135290003602464110665, 6.09597211860803420946915579794, 6.97396074818726247278344756109, 7.938479415372686610801448465611, 8.268819900668657794764803429742, 9.105281106982350730436816205338