| L(s) = 1 | + (1.17 + 1.61i)2-s + (−1.23 + 3.80i)4-s + 8.50·7-s + (−7.60 + 2.47i)8-s − 1.79i·11-s − 0.472i·13-s + (10 + 13.7i)14-s + (−12.9 − 9.40i)16-s + 23.8i·17-s + 9.40i·19-s + (2.90 − 2.11i)22-s + 16.1·23-s + (0.763 − 0.555i)26-s + (−10.5 + 32.3i)28-s + 6.94·29-s + ⋯ |
| L(s) = 1 | + (0.587 + 0.809i)2-s + (−0.309 + 0.951i)4-s + 1.21·7-s + (−0.951 + 0.309i)8-s − 0.163i·11-s − 0.0363i·13-s + (0.714 + 0.983i)14-s + (−0.809 − 0.587i)16-s + 1.40i·17-s + 0.494i·19-s + (0.132 − 0.0959i)22-s + 0.700·23-s + (0.0293 − 0.0213i)26-s + (−0.375 + 1.15i)28-s + 0.239·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.555387144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.555387144\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.17 - 1.61i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 8.50T + 49T^{2} \) |
| 11 | \( 1 + 1.79iT - 121T^{2} \) |
| 13 | \( 1 + 0.472iT - 169T^{2} \) |
| 17 | \( 1 - 23.8iT - 289T^{2} \) |
| 19 | \( 1 - 9.40iT - 361T^{2} \) |
| 23 | \( 1 - 16.1T + 529T^{2} \) |
| 29 | \( 1 - 6.94T + 841T^{2} \) |
| 31 | \( 1 - 47.4iT - 961T^{2} \) |
| 37 | \( 1 - 26.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 41.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 2.00T + 1.84e3T^{2} \) |
| 47 | \( 1 + 35.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 21.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 73.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 26.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 88.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 39.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 137. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 113. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 21.2T + 6.88e3T^{2} \) |
| 89 | \( 1 - 67.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 39.1iT - 9.40e3T^{2} \) |
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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
−10.34389381639009942824240687672, −9.007946298433079768090722577098, −8.334877051336607406830909892616, −7.74688324878581127706003995154, −6.74188007640968100829145379001, −5.84562710689147658989530964105, −4.98211340081565793079688743098, −4.19625424913180187149509191638, −3.07990334521480666752596870686, −1.54720723339663861218119057307,
0.71014231667832296768451325593, 1.99623585585548448141247441407, 2.97819924861037966761337921555, 4.33890603856692950220789682681, 4.91191942175595028591538102632, 5.80130448202558491961827201223, 7.00633433003882601833702257764, 7.927089058313221412679195967015, 9.050834342676205579469110473224, 9.626254045805853990293911829362
