| L(s) = 1 | + (−2.38 + 1.11i)3-s + (1.56 + 1.59i)5-s + (0.558 + 2.08i)7-s + (2.50 − 2.99i)9-s + (−0.730 − 1.26i)11-s + (−0.349 + 0.748i)13-s + (−5.49 − 2.07i)15-s + (−0.237 + 2.71i)17-s + (−4.25 + 0.959i)19-s + (−3.64 − 4.34i)21-s + (−5.66 + 3.96i)23-s + (−0.111 + 4.99i)25-s + (−0.614 + 2.29i)27-s + (−1.39 − 1.16i)29-s + (−1.26 − 0.729i)31-s + ⋯ |
| L(s) = 1 | + (−1.37 + 0.641i)3-s + (0.699 + 0.714i)5-s + (0.211 + 0.788i)7-s + (0.836 − 0.996i)9-s + (−0.220 − 0.381i)11-s + (−0.0968 + 0.207i)13-s + (−1.41 − 0.534i)15-s + (−0.0575 + 0.657i)17-s + (−0.975 + 0.220i)19-s + (−0.795 − 0.948i)21-s + (−1.18 + 0.827i)23-s + (−0.0223 + 0.999i)25-s + (−0.118 + 0.441i)27-s + (−0.258 − 0.217i)29-s + (−0.226 − 0.131i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.211373 + 0.677131i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.211373 + 0.677131i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.56 - 1.59i)T \) |
| 19 | \( 1 + (4.25 - 0.959i)T \) |
| good | 3 | \( 1 + (2.38 - 1.11i)T + (1.92 - 2.29i)T^{2} \) |
| 7 | \( 1 + (-0.558 - 2.08i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (0.730 + 1.26i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.349 - 0.748i)T + (-8.35 - 9.95i)T^{2} \) |
| 17 | \( 1 + (0.237 - 2.71i)T + (-16.7 - 2.95i)T^{2} \) |
| 23 | \( 1 + (5.66 - 3.96i)T + (7.86 - 21.6i)T^{2} \) |
| 29 | \( 1 + (1.39 + 1.16i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (1.26 + 0.729i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.309 - 0.309i)T + 37iT^{2} \) |
| 41 | \( 1 + (-0.339 + 0.931i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (5.64 - 8.06i)T + (-14.7 - 40.4i)T^{2} \) |
| 47 | \( 1 + (-10.9 + 0.955i)T + (46.2 - 8.16i)T^{2} \) |
| 53 | \( 1 + (5.47 + 7.82i)T + (-18.1 + 49.8i)T^{2} \) |
| 59 | \( 1 + (4.73 - 3.97i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.0111 + 0.0629i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (0.176 + 2.01i)T + (-65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (-13.0 + 2.29i)T + (66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-6.33 - 13.5i)T + (-46.9 + 55.9i)T^{2} \) |
| 79 | \( 1 + (-2.02 - 0.736i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-12.1 + 3.25i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-11.7 + 4.27i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (9.92 + 0.868i)T + (95.5 + 16.8i)T^{2} \) |
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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
−11.52274648904578598737130522544, −10.82141122549791955511284535279, −10.14468886052980250553858609939, −9.279091842939228237145275073388, −8.008582257029504059007337973747, −6.50229601399605120064561476022, −5.93095895189924717733846488785, −5.17909232950209587786894267815, −3.84187114791918766347125815599, −2.12420634543364050523739357956,
0.54616285806302582604546519684, 2.00652876727225268060282937356, 4.35346961407524333483803225765, 5.21153707627921864640307024482, 6.15275833211675456235659588896, 6.98550108431749576898850666278, 8.012002803864986462334196420762, 9.263655400637150724546083623947, 10.40924409814628225286322178651, 10.87172186802420155256815050813
