L(s) = 1 | + (−1.58 + 0.423i)2-s + (−1.71 + 0.990i)3-s + (0.587 − 0.339i)4-s + (−0.742 − 2.77i)5-s + (2.29 − 2.29i)6-s + (1.81 − 1.92i)7-s + (1.52 − 1.52i)8-s + (0.462 − 0.801i)9-s + (2.34 + 4.06i)10-s + (−3.33 − 0.894i)11-s + (−0.671 + 1.16i)12-s + (−2.29 − 2.78i)13-s + (−2.04 + 3.81i)14-s + (4.02 + 4.02i)15-s + (−2.44 + 4.24i)16-s + (−3.22 − 5.58i)17-s + ⋯ |
L(s) = 1 | + (−1.11 + 0.299i)2-s + (−0.990 + 0.571i)3-s + (0.293 − 0.169i)4-s + (−0.332 − 1.24i)5-s + (0.936 − 0.936i)6-s + (0.684 − 0.729i)7-s + (0.540 − 0.540i)8-s + (0.154 − 0.267i)9-s + (0.742 + 1.28i)10-s + (−1.00 − 0.269i)11-s + (−0.193 + 0.335i)12-s + (−0.635 − 0.771i)13-s + (−0.546 + 1.02i)14-s + (1.03 + 1.03i)15-s + (−0.612 + 1.06i)16-s + (−0.782 − 1.35i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0385 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0385 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.186985 - 0.179914i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.186985 - 0.179914i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-1.81 + 1.92i)T \) |
| 13 | \( 1 + (2.29 + 2.78i)T \) |
good | 2 | \( 1 + (1.58 - 0.423i)T + (1.73 - i)T^{2} \) |
| 3 | \( 1 + (1.71 - 0.990i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.742 + 2.77i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (3.33 + 0.894i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (3.22 + 5.58i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.786 - 2.93i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-2.97 - 1.71i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 4.40T + 29T^{2} \) |
| 31 | \( 1 + (-3.24 - 0.868i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (1.00 + 3.75i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-3.03 + 3.03i)T - 41iT^{2} \) |
| 43 | \( 1 - 4.48iT - 43T^{2} \) |
| 47 | \( 1 + (5.51 - 1.47i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-5.72 - 9.91i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.766 + 2.86i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (3.03 + 1.75i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.88 + 7.04i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-8.31 - 8.31i)T + 71iT^{2} \) |
| 73 | \( 1 + (-1.20 + 4.51i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (0.543 - 0.942i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.01 - 2.01i)T - 83iT^{2} \) |
| 89 | \( 1 + (-4.79 + 1.28i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-3.20 + 3.20i)T - 97iT^{2} \) |
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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
−13.68903990183899171990445073130, −12.65511156717211437792999915896, −11.31956579317497471708644571975, −10.49437709682851621950424318606, −9.450435478556155446987964155125, −8.216314214899960264459275619842, −7.43406146059490131283295428574, −5.29435697600451658573895785647, −4.53877031422089976007235046560, −0.50217844889111864523825724717,
2.23733696511419722037401966430, 5.03793447244144939790250479747, 6.58317445086193731157655750240, 7.61127585490560249247879319038, 8.829116422623790315979596918774, 10.28794016864142310552459158767, 11.13564964784143412717620331165, 11.66495527573975659357074638030, 13.02961968881660965442581303587, 14.57978091277577881519272756878