L(s) = 1 | + (1.16 − 0.804i)2-s + i·3-s + (0.707 − 1.87i)4-s + i·5-s + (0.804 + 1.16i)6-s − 4.03·7-s + (−0.681 − 2.74i)8-s − 9-s + (0.804 + 1.16i)10-s − 0.402·11-s + (1.87 + 0.707i)12-s + 0.442·13-s + (−4.69 + 3.24i)14-s − 15-s + (−2.99 − 2.64i)16-s − 4.35i·17-s + ⋯ |
L(s) = 1 | + (0.822 − 0.568i)2-s + 0.577i·3-s + (0.353 − 0.935i)4-s + 0.447i·5-s + (0.328 + 0.474i)6-s − 1.52·7-s + (−0.240 − 0.970i)8-s − 0.333·9-s + (0.254 + 0.367i)10-s − 0.121·11-s + (0.540 + 0.204i)12-s + 0.122·13-s + (−1.25 + 0.867i)14-s − 0.258·15-s + (−0.749 − 0.661i)16-s − 1.05i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7692266492\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7692266492\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.16 + 0.804i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (2.80 + 3.88i)T \) |
good | 7 | \( 1 + 4.03T + 7T^{2} \) |
| 11 | \( 1 + 0.402T + 11T^{2} \) |
| 13 | \( 1 - 0.442T + 13T^{2} \) |
| 17 | \( 1 + 4.35iT - 17T^{2} \) |
| 19 | \( 1 + 3.70T + 19T^{2} \) |
| 29 | \( 1 + 0.591T + 29T^{2} \) |
| 31 | \( 1 + 3.29iT - 31T^{2} \) |
| 37 | \( 1 + 0.0841iT - 37T^{2} \) |
| 41 | \( 1 + 9.40T + 41T^{2} \) |
| 43 | \( 1 + 0.681T + 43T^{2} \) |
| 47 | \( 1 + 5.37iT - 47T^{2} \) |
| 53 | \( 1 + 0.279iT - 53T^{2} \) |
| 59 | \( 1 - 12.4iT - 59T^{2} \) |
| 61 | \( 1 - 1.46iT - 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 3.55iT - 71T^{2} \) |
| 73 | \( 1 - 9.25T + 73T^{2} \) |
| 79 | \( 1 - 0.0927T + 79T^{2} \) |
| 83 | \( 1 - 3.36T + 83T^{2} \) |
| 89 | \( 1 - 5.03iT - 89T^{2} \) |
| 97 | \( 1 + 13.2iT - 97T^{2} \) |
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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
−9.533081611838617608514856102542, −8.722772767343131160979520010485, −7.26316631946838931776957941779, −6.47157791422778984347184819823, −5.89712750944745886201614685273, −4.81249809125784072174058241648, −3.88047178442751959606182506599, −3.12967344584108757042037646086, −2.32478384589005704505105031189, −0.21369233878907933729068463453,
1.91667940847937481587213498701, 3.19486066979372680884173116942, 3.87594556133916645472066743275, 5.05900285873150240304942758107, 6.11266200848243227073052931744, 6.39809344399459378291308443267, 7.33861552610782205374994492849, 8.214717600589421525713204846240, 8.888233870480739289684854319867, 9.872460058979799903208423436748