L(s) = 1 | + (−0.632 − 1.26i)2-s + 2.88·3-s + (−1.20 + 1.59i)4-s − 0.914·5-s + (−1.82 − 3.64i)6-s − 2.61i·7-s + (2.78 + 0.507i)8-s + 5.30·9-s + (0.578 + 1.15i)10-s − 0.394·11-s + (−3.45 + 4.60i)12-s + 3.33i·13-s + (−3.31 + 1.65i)14-s − 2.63·15-s + (−1.11 − 3.84i)16-s + (−2.66 − 3.14i)17-s + ⋯ |
L(s) = 1 | + (−0.447 − 0.894i)2-s + 1.66·3-s + (−0.600 + 0.799i)4-s − 0.408·5-s + (−0.743 − 1.48i)6-s − 0.990i·7-s + (0.983 + 0.179i)8-s + 1.76·9-s + (0.182 + 0.365i)10-s − 0.118·11-s + (−0.998 + 1.33i)12-s + 0.924i·13-s + (−0.885 + 0.442i)14-s − 0.680·15-s + (−0.279 − 0.960i)16-s + (−0.646 − 0.762i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.09659 - 0.633890i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09659 - 0.633890i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.632 + 1.26i)T \) |
| 17 | \( 1 + (2.66 + 3.14i)T \) |
good | 3 | \( 1 - 2.88T + 3T^{2} \) |
| 5 | \( 1 + 0.914T + 5T^{2} \) |
| 7 | \( 1 + 2.61iT - 7T^{2} \) |
| 11 | \( 1 + 0.394T + 11T^{2} \) |
| 13 | \( 1 - 3.33iT - 13T^{2} \) |
| 19 | \( 1 - 6.87iT - 19T^{2} \) |
| 23 | \( 1 + 0.692iT - 23T^{2} \) |
| 29 | \( 1 + 8.20T + 29T^{2} \) |
| 31 | \( 1 - 7.59iT - 31T^{2} \) |
| 37 | \( 1 - 8.98T + 37T^{2} \) |
| 41 | \( 1 + 6.90iT - 41T^{2} \) |
| 43 | \( 1 - 2.18iT - 43T^{2} \) |
| 47 | \( 1 - 5.96T + 47T^{2} \) |
| 53 | \( 1 + 7.24iT - 53T^{2} \) |
| 59 | \( 1 + 0.845iT - 59T^{2} \) |
| 61 | \( 1 + 3.22T + 61T^{2} \) |
| 67 | \( 1 + 8.21iT - 67T^{2} \) |
| 71 | \( 1 - 8.91iT - 71T^{2} \) |
| 73 | \( 1 + 12.9iT - 73T^{2} \) |
| 79 | \( 1 + 13.8iT - 79T^{2} \) |
| 83 | \( 1 - 5.82iT - 83T^{2} \) |
| 89 | \( 1 - 2.13T + 89T^{2} \) |
| 97 | \( 1 - 9.33iT - 97T^{2} \) |
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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
−13.22288132131477888728412547576, −12.04870959508621256823668758246, −10.83256185936646933730510412381, −9.765171849472749567898736998049, −8.998351052292834607256259216579, −7.924219143350227333676326783256, −7.25687652975821692099059621172, −4.26074857954080587841205606065, −3.51520336831531276305086729236, −1.94982426977383976531076896488,
2.45916562837243864806846332202, 4.17157002049387432638043544536, 5.82657606505942081516845350349, 7.40983713107493897404723600307, 8.139952772972788002513686016452, 8.990476670294366976946864906210, 9.665117286497868432238121087303, 11.15002068997204487107920825553, 12.97507093337054028410168942782, 13.46632832575282699807970809813