| L(s) = 1 | + (0.974 − 0.222i)2-s + (0.875 + 0.483i)3-s + (0.900 − 0.433i)4-s + (−0.656 − 1.18i)5-s + (0.960 + 0.276i)6-s + (1.12 + 1.41i)7-s + (0.781 − 0.623i)8-s + (0.532 + 0.846i)9-s + (−0.904 − 1.01i)10-s + (2.32 − 0.812i)11-s + (0.998 + 0.0560i)12-s + (0.363 − 3.22i)13-s + (1.41 + 1.12i)14-s − 1.35i·15-s + (0.623 − 0.781i)16-s + (0.348 − 0.247i)17-s + ⋯ |
| L(s) = 1 | + (0.689 − 0.157i)2-s + (0.505 + 0.279i)3-s + (0.450 − 0.216i)4-s + (−0.293 − 0.531i)5-s + (0.392 + 0.113i)6-s + (0.425 + 0.533i)7-s + (0.276 − 0.220i)8-s + (0.177 + 0.282i)9-s + (−0.286 − 0.320i)10-s + (0.699 − 0.244i)11-s + (0.288 + 0.0161i)12-s + (0.100 − 0.893i)13-s + (0.377 + 0.300i)14-s − 0.350i·15-s + (0.155 − 0.195i)16-s + (0.0845 − 0.0599i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 678 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 678 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.73667 - 0.361011i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.73667 - 0.361011i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.974 + 0.222i)T \) |
| 3 | \( 1 + (-0.875 - 0.483i)T \) |
| 113 | \( 1 + (5.70 + 8.96i)T \) |
| good | 5 | \( 1 + (0.656 + 1.18i)T + (-2.66 + 4.23i)T^{2} \) |
| 7 | \( 1 + (-1.12 - 1.41i)T + (-1.55 + 6.82i)T^{2} \) |
| 11 | \( 1 + (-2.32 + 0.812i)T + (8.60 - 6.85i)T^{2} \) |
| 13 | \( 1 + (-0.363 + 3.22i)T + (-12.6 - 2.89i)T^{2} \) |
| 17 | \( 1 + (-0.348 + 0.247i)T + (5.61 - 16.0i)T^{2} \) |
| 19 | \( 1 + (-1.95 + 0.564i)T + (16.0 - 10.1i)T^{2} \) |
| 23 | \( 1 + (2.32 - 8.07i)T + (-19.4 - 12.2i)T^{2} \) |
| 29 | \( 1 + (-1.08 - 1.52i)T + (-9.57 + 27.3i)T^{2} \) |
| 31 | \( 1 + (-4.87 - 0.549i)T + (30.2 + 6.89i)T^{2} \) |
| 37 | \( 1 + (1.66 - 1.49i)T + (4.14 - 36.7i)T^{2} \) |
| 41 | \( 1 + (8.65 + 3.02i)T + (32.0 + 25.5i)T^{2} \) |
| 43 | \( 1 + (-0.219 + 1.29i)T + (-40.5 - 14.2i)T^{2} \) |
| 47 | \( 1 + (10.4 - 0.585i)T + (46.7 - 5.26i)T^{2} \) |
| 53 | \( 1 + (1.08 + 2.25i)T + (-33.0 + 41.4i)T^{2} \) |
| 59 | \( 1 + (0.171 + 0.596i)T + (-49.9 + 31.3i)T^{2} \) |
| 61 | \( 1 + (-2.48 - 7.09i)T + (-47.6 + 38.0i)T^{2} \) |
| 67 | \( 1 + (0.618 - 11.0i)T + (-66.5 - 7.50i)T^{2} \) |
| 71 | \( 1 + (1.64 + 3.96i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (7.32 + 3.03i)T + (51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (2.92 - 3.27i)T + (-8.84 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-1.34 - 5.88i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (8.36 - 1.42i)T + (84.0 - 29.3i)T^{2} \) |
| 97 | \( 1 + (0.591 - 0.741i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−10.45350623917881101370821612013, −9.640404216570087964725933706407, −8.613162073206240258977811790063, −8.030633550443700660232108472690, −6.88114761544343810297405132369, −5.63860163807096806728710986170, −4.95316366558350717396506213531, −3.84642446222095820888089489865, −2.95356889198132700113290108660, −1.46234332857813554973808477356,
1.63335332292118793975198951713, 2.97995366491264812368000676374, 4.02417706350503628444698906406, 4.78201522694955058903096785311, 6.37208784813008451309166645569, 6.84588441080741146801064092248, 7.78123025758288919877277338076, 8.590523845882603518605424672322, 9.713096685163384415788446864706, 10.66039896381525623223269272841
