| L(s) = 1 | + (1.38 + 0.308i)2-s + (1.47 + 2.89i)3-s + (1.81 + 0.850i)4-s + (−2.23 − 0.158i)5-s + (1.14 + 4.45i)6-s + (0.707 + 0.707i)7-s + (2.23 + 1.73i)8-s + (−4.45 + 6.13i)9-s + (−3.02 − 0.906i)10-s + (−3.22 − 4.44i)11-s + (0.208 + 6.50i)12-s + (−0.209 − 1.32i)13-s + (0.758 + 1.19i)14-s + (−2.83 − 6.69i)15-s + (2.55 + 3.07i)16-s + (5.96 + 3.04i)17-s + ⋯ |
| L(s) = 1 | + (0.975 + 0.217i)2-s + (0.852 + 1.67i)3-s + (0.905 + 0.425i)4-s + (−0.997 − 0.0710i)5-s + (0.467 + 1.81i)6-s + (0.267 + 0.267i)7-s + (0.790 + 0.612i)8-s + (−1.48 + 2.04i)9-s + (−0.958 − 0.286i)10-s + (−0.972 − 1.33i)11-s + (0.0601 + 1.87i)12-s + (−0.0582 − 0.367i)13-s + (0.202 + 0.319i)14-s + (−0.731 − 1.72i)15-s + (0.638 + 0.769i)16-s + (1.44 + 0.737i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.35388 + 2.70846i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35388 + 2.70846i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.38 - 0.308i)T \) |
| 5 | \( 1 + (2.23 + 0.158i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| good | 3 | \( 1 + (-1.47 - 2.89i)T + (-1.76 + 2.42i)T^{2} \) |
| 11 | \( 1 + (3.22 + 4.44i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.209 + 1.32i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-5.96 - 3.04i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-1.18 - 3.64i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.486 + 3.07i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (2.15 + 0.700i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.45 + 0.471i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (3.43 - 0.544i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (0.264 + 0.192i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-5.96 + 5.96i)T - 43iT^{2} \) |
| 47 | \( 1 + (-6.64 + 3.38i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-6.58 + 3.35i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (1.54 + 1.12i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-10.2 + 7.46i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (4.30 - 8.44i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (14.4 + 4.68i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (5.72 + 0.906i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (1.34 - 4.15i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (4.95 + 2.52i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-5.99 - 8.24i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-7.13 - 14.0i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−10.59112052327437024218390861463, −10.30181159323055923541279017345, −8.724281343877217468359925210899, −8.193777074070379796999626705635, −7.61692094310419541929577959271, −5.68225086798067596366386030123, −5.26934375001950395745532955723, −4.06822773377861332436234498807, −3.49127840977174644292170347422, −2.72407647730164850128309054742,
1.15173315531403160533301175476, 2.45739900740404238453014868735, 3.23220149812162919775754013886, 4.49558789542695656610119937117, 5.64059687443337291681491221343, 7.07773560935321273085627772686, 7.34743540856148389992263322346, 7.83853234544900025853792242137, 9.144119380178146633630994648189, 10.32591402066578876883431195216
