L(s) = 1 | − 28.9·3-s + (−40.5 − 38.5i)5-s + 128. i·7-s + 592.·9-s + 433. i·11-s + 78.5·13-s + (1.17e3 + 1.11e3i)15-s + 1.48e3i·17-s − 98.3i·19-s − 3.70e3i·21-s − 2.36e3i·23-s + (156. + 3.12e3i)25-s − 1.01e4·27-s + 1.02e3i·29-s − 4.30e3·31-s + ⋯ |
L(s) = 1 | − 1.85·3-s + (−0.724 − 0.689i)5-s + 0.988i·7-s + 2.43·9-s + 1.08i·11-s + 0.128·13-s + (1.34 + 1.27i)15-s + 1.24i·17-s − 0.0624i·19-s − 1.83i·21-s − 0.931i·23-s + (0.0500 + 0.998i)25-s − 2.66·27-s + 0.226i·29-s − 0.805·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.09908897231\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09908897231\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (40.5 + 38.5i)T \) |
good | 3 | \( 1 + 28.9T + 243T^{2} \) |
| 7 | \( 1 - 128. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 433. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 78.5T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.48e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 98.3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 2.36e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 1.02e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 4.30e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.64e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.44e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.06e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.55e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.67e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.29e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 1.92e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 1.06e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.23e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.72e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 5.07e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.86e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.32e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.50e4iT - 8.58e9T^{2} \) |
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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
−11.75183686247578580386742735260, −10.88812340026141094017325439626, −9.822873832578092152267418930034, −8.507000970369879188861730392941, −7.15685155493541967927387849387, −6.04279474894988545967403080027, −5.09452140587285872118996145295, −4.19225269651197387629911723142, −1.60625939344933648187326834175, −0.05840398884547429569812112056,
0.912927682779105796797872245296, 3.53981516074750649122328512251, 4.74227617823142693879663796990, 5.96459017523859563931667680049, 6.93101231260824154959093974623, 7.72663812930570816063405685541, 9.701987807383711317585951437906, 10.80859492423929742162157906603, 11.22150091102777814618025129094, 11.92772566263234151443609573755