L(s) = 1 | + (0.939 − 0.342i)3-s + (0.939 − 1.62i)5-s + (0.766 + 1.32i)7-s + (0.766 − 0.642i)9-s + (−0.5 + 0.866i)13-s + (0.326 − 1.85i)15-s + 0.347·17-s + (1.17 + 0.984i)21-s + (−1.26 − 2.19i)25-s + (0.500 − 0.866i)27-s + (−0.5 + 0.866i)31-s + 2.87·35-s − 1.87·37-s + (−0.173 + 0.984i)39-s + (0.173 + 0.300i)43-s + ⋯ |
L(s) = 1 | + (0.939 − 0.342i)3-s + (0.939 − 1.62i)5-s + (0.766 + 1.32i)7-s + (0.766 − 0.642i)9-s + (−0.5 + 0.866i)13-s + (0.326 − 1.85i)15-s + 0.347·17-s + (1.17 + 0.984i)21-s + (−1.26 − 2.19i)25-s + (0.500 − 0.866i)27-s + (−0.5 + 0.866i)31-s + 2.87·35-s − 1.87·37-s + (−0.173 + 0.984i)39-s + (0.173 + 0.300i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.295638098\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.295638098\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.939 + 0.342i)T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - 0.347T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + 1.87T + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + 1.53T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−8.680908324548127657494778201423, −8.231425717918240144259201679795, −7.29019824380702548433440155366, −6.30795371785648381263106595352, −5.43364912949423072373234658277, −4.96442383802385424058142264513, −4.13290662533100953462873554894, −2.80068619899772810642715945836, −1.87183735885200913296928451193, −1.49827954681443338889068047044,
1.59438240604667677760719850812, 2.44357045637019992500120236814, 3.28881866872003456776600253792, 3.89686197769802497935337394068, 4.96712310917183544479371014632, 5.78432852953499541820823472338, 6.88186325582283943631328054631, 7.35324455429357595623416386600, 7.81105643176505738463898963639, 8.767349652447830330302247059532