L(s) = 1 | + (−0.555 − 0.831i)2-s + (−0.382 + 0.923i)4-s + (−0.980 − 0.195i)7-s + (0.980 − 0.195i)8-s + (−0.195 − 0.980i)9-s + (1.75 + 0.172i)11-s + (0.382 + 0.923i)14-s + (−0.707 − 0.707i)16-s + (−0.707 + 0.707i)18-s + (−0.831 − 1.55i)22-s + (−0.216 + 0.324i)23-s + (0.831 − 0.555i)25-s + (0.555 − 0.831i)28-s + (−0.187 − 1.90i)29-s + (−0.195 + 0.980i)32-s + ⋯ |
L(s) = 1 | + (−0.555 − 0.831i)2-s + (−0.382 + 0.923i)4-s + (−0.980 − 0.195i)7-s + (0.980 − 0.195i)8-s + (−0.195 − 0.980i)9-s + (1.75 + 0.172i)11-s + (0.382 + 0.923i)14-s + (−0.707 − 0.707i)16-s + (−0.707 + 0.707i)18-s + (−0.831 − 1.55i)22-s + (−0.216 + 0.324i)23-s + (0.831 − 0.555i)25-s + (0.555 − 0.831i)28-s + (−0.187 − 1.90i)29-s + (−0.195 + 0.980i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0490 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0490 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6955286195\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6955286195\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.555 + 0.831i)T \) |
| 7 | \( 1 + (0.980 + 0.195i)T \) |
good | 3 | \( 1 + (0.195 + 0.980i)T^{2} \) |
| 5 | \( 1 + (-0.831 + 0.555i)T^{2} \) |
| 11 | \( 1 + (-1.75 - 0.172i)T + (0.980 + 0.195i)T^{2} \) |
| 13 | \( 1 + (0.831 + 0.555i)T^{2} \) |
| 17 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 19 | \( 1 + (-0.555 + 0.831i)T^{2} \) |
| 23 | \( 1 + (0.216 - 0.324i)T + (-0.382 - 0.923i)T^{2} \) |
| 29 | \( 1 + (0.187 + 1.90i)T + (-0.980 + 0.195i)T^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 37 | \( 1 + (-0.938 + 1.75i)T + (-0.555 - 0.831i)T^{2} \) |
| 41 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 43 | \( 1 + (0.124 + 0.151i)T + (-0.195 + 0.980i)T^{2} \) |
| 47 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 53 | \( 1 + (0.151 - 1.53i)T + (-0.980 - 0.195i)T^{2} \) |
| 59 | \( 1 + (0.831 - 0.555i)T^{2} \) |
| 61 | \( 1 + (-0.195 - 0.980i)T^{2} \) |
| 67 | \( 1 + (0.980 + 0.804i)T + (0.195 + 0.980i)T^{2} \) |
| 71 | \( 1 + (0.360 - 1.81i)T + (-0.923 - 0.382i)T^{2} \) |
| 73 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 79 | \( 1 + (-1.53 - 0.636i)T + (0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 + (0.555 - 0.831i)T^{2} \) |
| 89 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.914094562086710966951659928656, −9.326871933226934922851924148855, −8.915855782081922388801307675172, −7.66809838621499521271509169673, −6.71290564449319489421469785339, −6.04230286152122709264183654730, −4.18279140816151804234520808952, −3.74793708021084738883022867779, −2.56039348865748451892362559962, −0.952617870113289477444173460893,
1.51033176330278371140291600663, 3.20781067310114762356564848131, 4.50731665360974250189918216592, 5.48871551992712429872346504254, 6.51930694494057068490421293663, 6.90311244073016788418898496734, 8.071607472927634567504091249977, 8.905668065034121038332442760689, 9.422067663688928956263704506723, 10.33653456149003417200359281843