L(s) = 1 | + (0.186 + 1.40i)2-s + (0.418 + 1.68i)3-s + (−1.93 + 0.523i)4-s + (1.60 − 2.78i)5-s + (−2.27 + 0.900i)6-s + (−1.82 + 1.05i)7-s + (−1.09 − 2.60i)8-s + (−2.64 + 1.40i)9-s + (4.20 + 1.73i)10-s + (3.47 − 2.00i)11-s + (−1.68 − 3.02i)12-s + (0.341 + 0.197i)13-s + (−1.81 − 2.35i)14-s + (5.35 + 1.53i)15-s + (3.45 − 2.02i)16-s − 1.20i·17-s + ⋯ |
L(s) = 1 | + (0.132 + 0.991i)2-s + (0.241 + 0.970i)3-s + (−0.965 + 0.261i)4-s + (0.719 − 1.24i)5-s + (−0.929 + 0.367i)6-s + (−0.688 + 0.397i)7-s + (−0.386 − 0.922i)8-s + (−0.883 + 0.469i)9-s + (1.33 + 0.548i)10-s + (1.04 − 0.605i)11-s + (−0.487 − 0.873i)12-s + (0.0948 + 0.0547i)13-s + (−0.485 − 0.630i)14-s + (1.38 + 0.397i)15-s + (0.862 − 0.505i)16-s − 0.292i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.698707 + 0.707759i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.698707 + 0.707759i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.186 - 1.40i)T \) |
| 3 | \( 1 + (-0.418 - 1.68i)T \) |
good | 5 | \( 1 + (-1.60 + 2.78i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (1.82 - 1.05i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.47 + 2.00i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.341 - 0.197i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 1.20iT - 17T^{2} \) |
| 19 | \( 1 + 1.62T + 19T^{2} \) |
| 23 | \( 1 + (2.74 - 4.75i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.95 + 5.12i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.34 + 1.93i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 10.8iT - 37T^{2} \) |
| 41 | \( 1 + (1.23 + 0.715i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.21 + 2.10i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.792 + 1.37i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 7.07T + 53T^{2} \) |
| 59 | \( 1 + (2.29 + 1.32i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-8.18 + 4.72i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.60 - 4.51i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2.69T + 71T^{2} \) |
| 73 | \( 1 - 9.49T + 73T^{2} \) |
| 79 | \( 1 + (1.53 - 0.886i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.30 - 0.755i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 11.2iT - 89T^{2} \) |
| 97 | \( 1 + (-5.84 - 10.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−15.11555088275243338813039169568, −13.89776642967543030649985482564, −13.19275989080261570340854398232, −11.81682419115256999962390995474, −9.751988621253314932949373698111, −9.225151376455100801530525660272, −8.318775129180915008105381154200, −6.20662500580077710268226757241, −5.25126877548436741070106933373, −3.84087732465043202255617100790,
2.10734416985607204939599599251, 3.58297790741058513013486155578, 6.09604931963359637617391915362, 7.08396045837329015440925329341, 8.921973091538967345697587913023, 10.06189489100026288681445055867, 11.05408796833931441802802494143, 12.34660958628558377417255351176, 13.15719534030385594029492997268, 14.31869720605212893287099243355