| L(s) = 1 | + (−3 − 5.19i)5-s + (18 − 31.1i)11-s − 62·13-s + (−57 + 98.7i)17-s + (−38 − 65.8i)19-s + (−12 − 20.7i)23-s + (44.5 − 77.0i)25-s − 54·29-s + (−56 + 96.9i)31-s + (89 + 154. i)37-s + 378·41-s − 172·43-s + (96 + 166. i)47-s + (−201 + 348. i)53-s − 216·55-s + ⋯ |
| L(s) = 1 | + (−0.268 − 0.464i)5-s + (0.493 − 0.854i)11-s − 1.32·13-s + (−0.813 + 1.40i)17-s + (−0.458 − 0.794i)19-s + (−0.108 − 0.188i)23-s + (0.355 − 0.616i)25-s − 0.345·29-s + (−0.324 + 0.561i)31-s + (0.395 + 0.684i)37-s + 1.43·41-s − 0.609·43-s + (0.297 + 0.516i)47-s + (−0.520 + 0.902i)53-s − 0.529·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.154658353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.154658353\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (3 + 5.19i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-18 + 31.1i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 62T + 2.19e3T^{2} \) |
| 17 | \( 1 + (57 - 98.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (38 + 65.8i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (12 + 20.7i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 54T + 2.43e4T^{2} \) |
| 31 | \( 1 + (56 - 96.9i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-89 - 154. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 378T + 6.89e4T^{2} \) |
| 43 | \( 1 + 172T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-96 - 166. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (201 - 348. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (198 - 342. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-127 - 219. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-506 + 876. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 840T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-445 + 770. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (40 + 69.2i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 108T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-819 - 1.41e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.01e3T + 9.12e5T^{2} \) |
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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
−8.923497652162725536121609070862, −8.377416197079429333854448799420, −7.52185868061613281143560104556, −6.57334375772520202614523992490, −5.91861075097303946200358644003, −4.74411503840821895583870477903, −4.23843192129614131371634627072, −3.06001763876551630952708410185, −2.01840154608359979620468136520, −0.73809328277203389995923891327,
0.33109818455107951230742245447, 1.92024308705085918366006901710, 2.70456784454764596783976509523, 3.88302496441562025885601956134, 4.67221805095848334298301244823, 5.52363707822920626694777538804, 6.69320365203738603565965598293, 7.23050638232465351042308915128, 7.82001890065565460814948374560, 9.029255558592590997349599013697
