L(s) = 1 | + (−0.569 − 1.29i)2-s + (−1.47 + 0.908i)3-s + (−1.35 + 1.47i)4-s + (2.01 + 1.39i)6-s − 2.50·7-s + (2.67 + 0.908i)8-s + (1.35 − 2.67i)9-s + 3.36·11-s + (0.652 − 3.40i)12-s − 3.70i·13-s + (1.42 + 3.24i)14-s + (−0.350 − 3.98i)16-s + 7.63·17-s + (−4.23 − 0.222i)18-s − 0.440i·19-s + ⋯ |
L(s) = 1 | + (−0.402 − 0.915i)2-s + (−0.851 + 0.524i)3-s + (−0.675 + 0.737i)4-s + (0.822 + 0.568i)6-s − 0.948·7-s + (0.947 + 0.321i)8-s + (0.450 − 0.892i)9-s + 1.01·11-s + (0.188 − 0.982i)12-s − 1.02i·13-s + (0.382 + 0.868i)14-s + (−0.0876 − 0.996i)16-s + 1.85·17-s + (−0.998 − 0.0523i)18-s − 0.100i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.555943 - 0.421217i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.555943 - 0.421217i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.569 + 1.29i)T \) |
| 3 | \( 1 + (1.47 - 0.908i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2.50T + 7T^{2} \) |
| 11 | \( 1 - 3.36T + 11T^{2} \) |
| 13 | \( 1 + 3.70iT - 13T^{2} \) |
| 17 | \( 1 - 7.63T + 17T^{2} \) |
| 19 | \( 1 + 0.440iT - 19T^{2} \) |
| 23 | \( 1 + 5.17iT - 23T^{2} \) |
| 29 | \( 1 + 2.27iT - 29T^{2} \) |
| 31 | \( 1 + 3.39iT - 31T^{2} \) |
| 37 | \( 1 - 7.40iT - 37T^{2} \) |
| 41 | \( 1 + 3.07iT - 41T^{2} \) |
| 43 | \( 1 - 8.40T + 43T^{2} \) |
| 47 | \( 1 + 3.63iT - 47T^{2} \) |
| 53 | \( 1 - 2.27T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 5.70T + 61T^{2} \) |
| 67 | \( 1 - 5.45T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 1.29iT - 73T^{2} \) |
| 79 | \( 1 + 5.01iT - 79T^{2} \) |
| 83 | \( 1 - 1.81iT - 83T^{2} \) |
| 89 | \( 1 + 5.35iT - 89T^{2} \) |
| 97 | \( 1 + 11.1iT - 97T^{2} \) |
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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
−11.55368446010273638253829058870, −10.44603914195864001618195762273, −9.937079738008680024089467349384, −9.156361441753445605849659994146, −7.87242266247241563547758425359, −6.53420561691995709174069837783, −5.41766811318096383045740179730, −4.04987960885708351416749112269, −3.11516937366209048125128476560, −0.805415195513620145705878779717,
1.28747247565519374857029627366, 3.87810877589539292827079406723, 5.33855979467151963496962714324, 6.19979834866345022101581883709, 6.96239254603095396822197135990, 7.78825465005750226087445033399, 9.216741945695917161450370745414, 9.807493567342501160732574988252, 10.93942456838834620271854315117, 12.04070696431940099206099793967