L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.423 − 1.67i)3-s + (0.499 − 0.866i)4-s + 5-s + (1.20 + 1.24i)6-s + (2.20 + 1.46i)7-s + 0.999i·8-s + (−2.64 + 1.42i)9-s + (−0.866 + 0.5i)10-s + 4.43i·11-s + (−1.66 − 0.472i)12-s + (−4.51 + 2.60i)13-s + (−2.64 − 0.172i)14-s + (−0.423 − 1.67i)15-s + (−0.5 − 0.866i)16-s + (3.46 + 6.00i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.244 − 0.969i)3-s + (0.249 − 0.433i)4-s + 0.447·5-s + (0.492 + 0.507i)6-s + (0.831 + 0.555i)7-s + 0.353i·8-s + (−0.880 + 0.474i)9-s + (−0.273 + 0.158i)10-s + 1.33i·11-s + (−0.480 − 0.136i)12-s + (−1.25 + 0.723i)13-s + (−0.705 − 0.0460i)14-s + (−0.109 − 0.433i)15-s + (−0.125 − 0.216i)16-s + (0.841 + 1.45i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.777574 + 0.517698i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.777574 + 0.517698i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.423 + 1.67i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-2.20 - 1.46i)T \) |
good | 11 | \( 1 - 4.43iT - 11T^{2} \) |
| 13 | \( 1 + (4.51 - 2.60i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.46 - 6.00i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.43 + 1.40i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 3.48iT - 23T^{2} \) |
| 29 | \( 1 + (6.40 + 3.69i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.82 + 1.05i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.36 + 7.56i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.90 - 5.02i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.02 - 1.77i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.16 - 8.95i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.60 - 2.65i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.534 - 0.926i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.23 + 3.02i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.46 - 4.26i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 11.5iT - 71T^{2} \) |
| 73 | \( 1 + (-13.1 + 7.61i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.88 + 6.72i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.84 - 4.93i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.16 + 7.22i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (11.4 + 6.62i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.77686271202057277870587840226, −9.678036881860866119945805643386, −9.044358751588863582921832349223, −7.69570218159298640128236171742, −7.59875763915418521329669953069, −6.33997767158169257276302624672, −5.58316960336219785300495596909, −4.54447121982416597924996420871, −2.25791607644516992305458166944, −1.69899417507400513821348653317,
0.63905316251421440274480905776, 2.59034660872798555341058412003, 3.65070682423994849646093232227, 4.96847979405238825030522275087, 5.61960825224644049708317629916, 7.03703038627217391510175476323, 8.049388704937328238030015770435, 8.833797720928196961745345485423, 9.780453773408689533147450249114, 10.35365720975027289932469178927