L(s) = 1 | + (−0.923 + 0.382i)2-s + (0.707 − 0.707i)4-s + (0.324 − 0.216i)5-s + (−0.382 + 0.923i)8-s + (−0.923 − 0.382i)9-s + (−0.216 + 0.324i)10-s + (−1 + i)13-s − i·16-s + (0.382 + 0.923i)17-s + 18-s + (0.0761 − 0.382i)20-s + (−0.324 + 0.783i)25-s + (0.541 − 1.30i)26-s + (−0.324 + 0.216i)29-s + (0.382 + 0.923i)32-s + ⋯ |
L(s) = 1 | + (−0.923 + 0.382i)2-s + (0.707 − 0.707i)4-s + (0.324 − 0.216i)5-s + (−0.382 + 0.923i)8-s + (−0.923 − 0.382i)9-s + (−0.216 + 0.324i)10-s + (−1 + i)13-s − i·16-s + (0.382 + 0.923i)17-s + 18-s + (0.0761 − 0.382i)20-s + (−0.324 + 0.783i)25-s + (0.541 − 1.30i)26-s + (−0.324 + 0.216i)29-s + (0.382 + 0.923i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.570 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.570 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4570380359\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4570380359\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.923 - 0.382i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (-0.382 - 0.923i)T \) |
good | 3 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 5 | \( 1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)T^{2} \) |
| 11 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 13 | \( 1 + (1 - i)T - iT^{2} \) |
| 19 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 29 | \( 1 + (0.324 - 0.216i)T + (0.382 - 0.923i)T^{2} \) |
| 31 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 37 | \( 1 + (1.92 + 0.382i)T + (0.923 + 0.382i)T^{2} \) |
| 41 | \( 1 + (-1.38 - 0.923i)T + (0.382 + 0.923i)T^{2} \) |
| 43 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (0.617 - 0.923i)T + (-0.382 - 0.923i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 73 | \( 1 + (1.63 - 1.08i)T + (0.382 - 0.923i)T^{2} \) |
| 79 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 83 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 + (-1.08 - 1.63i)T + (-0.382 + 0.923i)T^{2} \) |
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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
−9.076546363754481332859427693114, −8.461068250966191666386714341940, −7.58819793376078127658295681260, −6.98015004168892720751281188725, −6.07385649387455849445724317849, −5.59319404863814753206424202650, −4.65732308723604154785255643209, −3.40537319376834261299344118881, −2.33651516849284271809292187031, −1.43757215197905116700486468789,
0.35169130766545429036568927327, 1.98546723564408394121380620457, 2.75131713674030097519448378644, 3.40630547501026447624257053266, 4.79125151442604023245042200566, 5.61858089353086012755213700494, 6.39564889153704908392412907399, 7.45449487662231679895942417661, 7.73541175356816709009287910831, 8.674204149084784285894324916276