| L(s) = 1 | − 1.41i·2-s + (−2.58 + 1.52i)3-s − 2.00·4-s + (2.16 + 3.65i)6-s + 7.48·7-s + 2.82i·8-s + (4.32 − 7.89i)9-s − 8.48i·11-s + (5.16 − 3.05i)12-s + 10·13-s − 10.5i·14-s + 4.00·16-s − 30.3i·17-s + (−11.1 − 6.11i)18-s + 26.9·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (−0.860 + 0.509i)3-s − 0.500·4-s + (0.360 + 0.608i)6-s + 1.06·7-s + 0.353i·8-s + (0.480 − 0.876i)9-s − 0.771i·11-s + (0.430 − 0.254i)12-s + 0.769·13-s − 0.756i·14-s + 0.250·16-s − 1.78i·17-s + (−0.620 − 0.339i)18-s + 1.41·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.02180 - 0.582343i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02180 - 0.582343i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (2.58 - 1.52i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 7.48T + 49T^{2} \) |
| 11 | \( 1 + 8.48iT - 121T^{2} \) |
| 13 | \( 1 - 10T + 169T^{2} \) |
| 17 | \( 1 + 30.3iT - 289T^{2} \) |
| 19 | \( 1 - 26.9T + 361T^{2} \) |
| 23 | \( 1 - 9.17iT - 529T^{2} \) |
| 29 | \( 1 - 26.8iT - 841T^{2} \) |
| 31 | \( 1 - 8T + 961T^{2} \) |
| 37 | \( 1 + 15.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 47.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 14.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 45.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 30.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 24.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 53.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 110.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 15.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 87.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 46.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 26.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 60.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 36.0T + 9.40e3T^{2} \) |
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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
−12.19511647579530498777390672300, −11.39351963583036740819648516960, −10.97427136854200822382902541106, −9.724603358758645547544273593474, −8.761096473652082846573651735117, −7.31402037150613291820982282535, −5.62508244741687529326177114232, −4.81004843455721341772227440068, −3.33881546726969103933415846053, −1.04011146858540532785026418240,
1.49475741540889836105115585960, 4.29589653168940889926659910040, 5.42782359261958820366836329993, 6.43734244424244930160690884568, 7.63181985783494399171579236721, 8.376872684573774713276035738475, 9.986132971171234368036596227746, 11.03284582182149795459372153246, 11.98722925369306775940975951950, 12.97258184988897736013397746982
