L(s) = 1 | + (0.856 + 1.48i)2-s + (−3.79 − 6.56i)3-s + (2.53 − 4.38i)4-s + 12.0·5-s + (6.49 − 11.2i)6-s + (−0.574 + 0.994i)7-s + 22.3·8-s + (−15.2 + 26.4i)9-s + (10.3 + 17.8i)10-s + (−5.5 − 9.52i)11-s − 38.3·12-s + (−34.0 − 32.2i)13-s − 1.96·14-s + (−45.5 − 78.9i)15-s + (−1.06 − 1.84i)16-s + (6.74 − 11.6i)17-s + ⋯ |
L(s) = 1 | + (0.302 + 0.524i)2-s + (−0.729 − 1.26i)3-s + (0.316 − 0.548i)4-s + 1.07·5-s + (0.442 − 0.765i)6-s + (−0.0310 + 0.0537i)7-s + 0.989·8-s + (−0.564 + 0.977i)9-s + (0.325 + 0.564i)10-s + (−0.150 − 0.261i)11-s − 0.923·12-s + (−0.725 − 0.687i)13-s − 0.0375·14-s + (−0.784 − 1.35i)15-s + (−0.0166 − 0.0288i)16-s + (0.0962 − 0.166i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0334 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0334 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.35926 - 1.31459i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.35926 - 1.31459i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (5.5 + 9.52i)T \) |
| 13 | \( 1 + (34.0 + 32.2i)T \) |
good | 2 | \( 1 + (-0.856 - 1.48i)T + (-4 + 6.92i)T^{2} \) |
| 3 | \( 1 + (3.79 + 6.56i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 - 12.0T + 125T^{2} \) |
| 7 | \( 1 + (0.574 - 0.994i)T + (-171.5 - 297. i)T^{2} \) |
| 17 | \( 1 + (-6.74 + 11.6i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-48.2 + 83.6i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-32.7 - 56.7i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-50.3 - 87.2i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 259.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-25.9 - 44.8i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-102. - 177. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-209. + 363. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 183.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 183.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-369. + 640. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (310. - 537. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-397. - 688. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (208. - 361. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + 231.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 141.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-636. - 1.10e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (105. - 182. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−12.65711294233736976175657392856, −11.45798725211607797418781920547, −10.50614489609645040155863038459, −9.359363460872221373193038152319, −7.53928365628520556520730665673, −6.87660442753729877470492613604, −5.74721128681246520297795125115, −5.28377205723561147344608558267, −2.31978490034379194487407957988, −0.953111678265519893417878681464,
2.14112385453364722213256004040, 3.79429597387805746777986675335, 4.89318917592312296315573714942, 6.00557394452374959911535942025, 7.48070583425871045967865734110, 9.235985674494628546031507868967, 10.07403373168679711885938371149, 10.77948542543754915218798729596, 11.79560946964764438594024572939, 12.65618585422182438244386099876