| L(s) = 1 | + (−2.19 + 1.61i)2-s + (−0.278 + 1.47i)3-s + (1.59 − 5.17i)4-s + (−1.91 + 1.14i)5-s + (−1.77 − 3.67i)6-s + (−1.90 − 1.83i)7-s + (3.07 + 8.78i)8-s + (0.700 + 0.274i)9-s + (2.34 − 5.61i)10-s + (−1.69 − 4.31i)11-s + (7.18 + 3.79i)12-s + (−0.382 − 3.39i)13-s + (7.14 + 0.930i)14-s + (−1.15 − 3.14i)15-s + (−11.9 − 8.17i)16-s + (−0.0192 − 0.515i)17-s + ⋯ |
| L(s) = 1 | + (−1.54 + 1.14i)2-s + (−0.160 + 0.850i)3-s + (0.798 − 2.58i)4-s + (−0.858 + 0.513i)5-s + (−0.723 − 1.50i)6-s + (−0.721 − 0.692i)7-s + (1.08 + 3.10i)8-s + (0.233 + 0.0916i)9-s + (0.742 − 1.77i)10-s + (−0.510 − 1.30i)11-s + (2.07 + 1.09i)12-s + (−0.105 − 0.940i)13-s + (1.90 + 0.248i)14-s + (−0.298 − 0.812i)15-s + (−2.99 − 2.04i)16-s + (−0.00467 − 0.124i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.257735 - 0.0326450i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.257735 - 0.0326450i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.91 - 1.14i)T \) |
| 7 | \( 1 + (1.90 + 1.83i)T \) |
| good | 2 | \( 1 + (2.19 - 1.61i)T + (0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (0.278 - 1.47i)T + (-2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (1.69 + 4.31i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (0.382 + 3.39i)T + (-12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (0.0192 + 0.515i)T + (-16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (-1.16 - 2.01i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.23 - 0.120i)T + (22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (0.821 + 0.187i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (0.940 + 0.543i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.05 + 7.67i)T + (-20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (0.358 - 0.744i)T + (-25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (3.75 + 1.31i)T + (33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (7.35 + 9.96i)T + (-13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (0.849 + 1.60i)T + (-29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (-0.338 + 4.51i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (0.194 + 0.629i)T + (-50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (2.84 + 10.6i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (3.21 + 14.0i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (0.347 - 0.471i)T + (-21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (-1.16 + 0.670i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (14.4 + 1.63i)T + (80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (-2.10 + 5.35i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + (-3.98 - 3.98i)T + 97iT^{2} \) |
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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
−11.31517835834492085154156340246, −10.59962142992153564205918611193, −10.16191372810032374994239118158, −9.115917126311850533118955256517, −7.998007152764041955845864850308, −7.38141517991387453655691248502, −6.28839402748867809441316420557, −5.15518621564148580030968009094, −3.41505664262060329108206684042, −0.36624129335175887332965057761,
1.46341961626981008081110406842, 2.80843112089948842956974329768, 4.37996156923521627532497686369, 6.80899599764110676062121770097, 7.39984048934650793735746584476, 8.409613510324420404856621791628, 9.353477769238092436927953032308, 9.993771459168295951849793805352, 11.42222580940816810605198851293, 11.90503722353964436645564822379
