L(s) = 1 | + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s + (−2.56 − 0.667i)7-s + (−0.499 + 0.866i)9-s + (2.65 + 4.59i)11-s − 5.71·13-s − 0.999·15-s + (−3.76 − 6.51i)17-s + (−0.5 + 0.866i)19-s + (0.702 + 2.55i)21-s + (−2.06 + 3.56i)23-s + (−0.499 − 0.866i)25-s + 0.999·27-s − 2.71·29-s + (2.91 + 5.05i)31-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.223 − 0.387i)5-s + (−0.967 − 0.252i)7-s + (−0.166 + 0.288i)9-s + (0.800 + 1.38i)11-s − 1.58·13-s − 0.258·15-s + (−0.912 − 1.58i)17-s + (−0.114 + 0.198i)19-s + (0.153 + 0.556i)21-s + (−0.429 + 0.744i)23-s + (−0.0999 − 0.173i)25-s + 0.192·27-s − 0.504·29-s + (0.524 + 0.907i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.754 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0385716 + 0.103225i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0385716 + 0.103225i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.56 + 0.667i)T \) |
good | 11 | \( 1 + (-2.65 - 4.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5.71T + 13T^{2} \) |
| 17 | \( 1 + (3.76 + 6.51i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.06 - 3.56i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2.71T + 29T^{2} \) |
| 31 | \( 1 + (-2.91 - 5.05i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.45 - 4.24i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 8.12T + 41T^{2} \) |
| 43 | \( 1 - 4.31T + 43T^{2} \) |
| 47 | \( 1 + (5.41 - 9.38i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.29 - 3.97i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (7.35 + 12.7i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.15 + 3.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + (0.439 + 0.761i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.51 + 6.08i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 2.71T + 83T^{2} \) |
| 89 | \( 1 + (1.95 - 3.38i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−10.30456256774841418199699824781, −9.570329878371942001968697057976, −9.191348364562134939966604940453, −7.68621601082916476692195358188, −7.02188006157482575888805549218, −6.45498405395705327494953657176, −5.10742875381043783037825335929, −4.45876845610637734333350449876, −2.92310045113326080228739676301, −1.75022989331057447238414283647,
0.05182487095869677287064870680, 2.29689182130536587221841021023, 3.43860768930661203081683556792, 4.32045508774351851217017731541, 5.68263427510198643044627622021, 6.27403310719250508365421778685, 7.04899702566955433792047668490, 8.441691598061860833622318331783, 9.082512607707235917772713000760, 10.00606491512870783782227111022