L(s) = 1 | − 3-s + 5-s + (1.07 − 1.86i)7-s + 9-s + (2.31 − 4.00i)11-s + (2.99 + 5.19i)13-s − 15-s + (3.06 + 5.31i)17-s + (2.62 + 4.54i)19-s + (−1.07 + 1.86i)21-s + (−0.436 − 0.756i)23-s + 25-s − 27-s + (−2.60 + 4.50i)29-s + (1.56 − 2.71i)31-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + (0.407 − 0.705i)7-s + 0.333·9-s + (0.697 − 1.20i)11-s + (0.831 + 1.44i)13-s − 0.258·15-s + (0.743 + 1.28i)17-s + (0.601 + 1.04i)19-s + (−0.235 + 0.407i)21-s + (−0.0910 − 0.157i)23-s + 0.200·25-s − 0.192·27-s + (−0.483 + 0.837i)29-s + (0.281 − 0.488i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.962269910\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.962269910\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (-3.47 + 7.41i)T \) |
good | 7 | \( 1 + (-1.07 + 1.86i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.31 + 4.00i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.99 - 5.19i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.06 - 5.31i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.62 - 4.54i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.436 + 0.756i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.60 - 4.50i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.56 + 2.71i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.89 - 3.28i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.79 - 8.30i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 5.46T + 43T^{2} \) |
| 47 | \( 1 + (3.90 - 6.75i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 6.77T + 53T^{2} \) |
| 59 | \( 1 - 5.61T + 59T^{2} \) |
| 61 | \( 1 + (1.75 + 3.03i)T + (-30.5 + 52.8i)T^{2} \) |
| 71 | \( 1 + (-2.58 + 4.48i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.841 + 1.45i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.70 - 4.68i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.83 - 13.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 7.97T + 89T^{2} \) |
| 97 | \( 1 + (-5.01 - 8.68i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.345901909797083754156807745083, −7.984144919234325824828164233739, −6.78949692761478991526927127462, −6.28396751377137535370372980475, −5.77054936541997273453720293279, −4.75977250267261982160647398789, −3.89281546289267403468870826280, −3.35888878014467608075248048856, −1.56155096537628912344190344109, −1.26604483214766713595862910451,
0.68410962984173724828145689778, 1.77888440389977831971128789591, 2.77523677596611259481750597491, 3.74894920511816633912188637560, 4.96975601016880334998068842798, 5.26991382009119725256481417647, 6.01730262778957795213661418936, 6.94800360022692747333258308407, 7.46848176632018880203381932760, 8.415070758139577926737612102300