| L(s) = 1 | + (−0.996 − 0.0871i)2-s + (1.33 − 1.10i)3-s + (0.984 + 0.173i)4-s + (2.23 + 0.0174i)5-s + (−1.42 + 0.986i)6-s + (1.90 + 1.33i)7-s + (−0.965 − 0.258i)8-s + (0.548 − 2.94i)9-s + (−2.22 − 0.212i)10-s + (−0.889 + 2.44i)11-s + (1.50 − 0.859i)12-s + (0.203 + 2.32i)13-s + (−1.78 − 1.49i)14-s + (2.99 − 2.45i)15-s + (0.939 + 0.342i)16-s + (−5.64 + 1.51i)17-s + ⋯ |
| L(s) = 1 | + (−0.704 − 0.0616i)2-s + (0.768 − 0.639i)3-s + (0.492 + 0.0868i)4-s + (0.999 + 0.00779i)5-s + (−0.581 + 0.402i)6-s + (0.721 + 0.505i)7-s + (−0.341 − 0.0915i)8-s + (0.182 − 0.983i)9-s + (−0.703 − 0.0671i)10-s + (−0.268 + 0.736i)11-s + (0.434 − 0.248i)12-s + (0.0563 + 0.644i)13-s + (−0.477 − 0.400i)14-s + (0.773 − 0.633i)15-s + (0.234 + 0.0855i)16-s + (−1.36 + 0.366i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.37542 - 0.300170i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37542 - 0.300170i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.996 + 0.0871i)T \) |
| 3 | \( 1 + (-1.33 + 1.10i)T \) |
| 5 | \( 1 + (-2.23 - 0.0174i)T \) |
| good | 7 | \( 1 + (-1.90 - 1.33i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (0.889 - 2.44i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.203 - 2.32i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (5.64 - 1.51i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-2.11 + 1.21i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.43 + 7.76i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-5.39 + 4.52i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.28 - 7.30i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.48 + 5.55i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (1.75 - 2.09i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.950 + 2.03i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (4.66 - 6.65i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-1.68 + 1.68i)T - 53iT^{2} \) |
| 59 | \( 1 + (5.56 - 2.02i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.241 + 1.36i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (11.1 - 0.976i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (1.67 + 0.967i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.894 - 3.33i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-2.76 - 3.29i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (1.24 - 14.2i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-4.56 - 7.90i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.02 + 2.80i)T + (62.3 - 74.3i)T^{2} \) |
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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
−11.96197498360373306277777854976, −10.72377066915793518341166048558, −9.755772469837315668414190864818, −8.854684800568623656011170661654, −8.272917706929265331446898493381, −6.98219811563362648705793588175, −6.23108569267103120860325802940, −4.59779853235624810786906834057, −2.50825782410757945700157502055, −1.79183157155241173679023164367,
1.79708640452319909202713647416, 3.19324270545610888877330830058, 4.81647740674056833642846186635, 5.95063433154278114330625293376, 7.42986482215501153746986562369, 8.295387253839843736090591479409, 9.144762358452683667340450639927, 10.03935472824228202990515349854, 10.68861491353338652457391743271, 11.60003444488303032761010801265
