L(s) = 1 | + 1.41i·2-s + 2.07·3-s − 2.00·4-s + (4.65 − 1.82i)5-s + 2.93i·6-s + (3.36 + 6.13i)7-s − 2.82i·8-s − 4.67·9-s + (2.57 + 6.58i)10-s + 1.67·11-s − 4.15·12-s − 9.81·13-s + (−8.67 + 4.76i)14-s + (9.67 − 3.78i)15-s + 4.00·16-s + 0.498·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.692·3-s − 0.500·4-s + (0.931 − 0.364i)5-s + 0.489i·6-s + (0.481 + 0.876i)7-s − 0.353i·8-s − 0.519·9-s + (0.257 + 0.658i)10-s + 0.152·11-s − 0.346·12-s − 0.754·13-s + (−0.619 + 0.340i)14-s + (0.645 − 0.252i)15-s + 0.250·16-s + 0.0293·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.641 - 0.767i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.641 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.40561 + 0.657396i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.40561 + 0.657396i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 5 | \( 1 + (-4.65 + 1.82i)T \) |
| 7 | \( 1 + (-3.36 - 6.13i)T \) |
good | 3 | \( 1 - 2.07T + 9T^{2} \) |
| 11 | \( 1 - 1.67T + 121T^{2} \) |
| 13 | \( 1 + 9.81T + 169T^{2} \) |
| 17 | \( 1 - 0.498T + 289T^{2} \) |
| 19 | \( 1 + 32.2iT - 361T^{2} \) |
| 23 | \( 1 - 3.78iT - 529T^{2} \) |
| 29 | \( 1 + 39.0T + 841T^{2} \) |
| 31 | \( 1 + 24.9iT - 961T^{2} \) |
| 37 | \( 1 + 16.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 71.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 37.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 71.4T + 2.20e3T^{2} \) |
| 53 | \( 1 - 83.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 36.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 50.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 12.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 44.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 102.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 132.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 8.72T + 6.88e3T^{2} \) |
| 89 | \( 1 - 54.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 41.7T + 9.40e3T^{2} \) |
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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
−14.67418192377558264492906094726, −13.75705033959753691405643579356, −12.78842621609547949865086423491, −11.32973728348948688310539493283, −9.443892555904960533253243179000, −8.972885549873167082606662465635, −7.70082180835934991328617152505, −6.04103687321904440925773429341, −4.91915005322869435039504298154, −2.49387070979163103028626077794,
2.01786250826721560595192950375, 3.65993553966271882599333184392, 5.52901788855509461013281414676, 7.39110606537121890699172019303, 8.740574036838711160546761124114, 9.928949896571687439190084653003, 10.74350834793273714484505264604, 12.10782437379864147610685395965, 13.43890985889489058056765723055, 14.22966410967355446965436699232