L(s) = 1 | + (0.189 + 0.707i)2-s + (1.26 − 0.731i)4-s + (1.23 + 1.23i)5-s + (−2.64 − 0.0736i)7-s + (1.79 + 1.79i)8-s + (−0.639 + 1.10i)10-s + (3.18 − 0.854i)11-s + (2.53 − 2.56i)13-s + (−0.449 − 1.88i)14-s + (0.532 − 0.923i)16-s + (0.433 + 0.751i)17-s + (−1.01 + 3.80i)19-s + (2.46 + 0.660i)20-s + (1.21 + 2.09i)22-s + (3.77 + 2.17i)23-s + ⋯ |
L(s) = 1 | + (0.134 + 0.500i)2-s + (0.633 − 0.365i)4-s + (0.551 + 0.551i)5-s + (−0.999 − 0.0278i)7-s + (0.634 + 0.634i)8-s + (−0.202 + 0.350i)10-s + (0.961 − 0.257i)11-s + (0.701 − 0.712i)13-s + (−0.120 − 0.504i)14-s + (0.133 − 0.230i)16-s + (0.105 + 0.182i)17-s + (−0.233 + 0.872i)19-s + (0.551 + 0.147i)20-s + (0.257 + 0.446i)22-s + (0.786 + 0.454i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.04061 + 0.775629i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.04061 + 0.775629i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.64 + 0.0736i)T \) |
| 13 | \( 1 + (-2.53 + 2.56i)T \) |
good | 2 | \( 1 + (-0.189 - 0.707i)T + (-1.73 + i)T^{2} \) |
| 5 | \( 1 + (-1.23 - 1.23i)T + 5iT^{2} \) |
| 11 | \( 1 + (-3.18 + 0.854i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-0.433 - 0.751i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.01 - 3.80i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-3.77 - 2.17i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.65 - 4.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.220 - 0.220i)T + 31iT^{2} \) |
| 37 | \( 1 + (3.57 - 0.957i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (1.90 - 0.509i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-9.99 + 5.76i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.68 - 3.68i)T - 47iT^{2} \) |
| 53 | \( 1 - 3.55T + 53T^{2} \) |
| 59 | \( 1 + (8.89 + 2.38i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-4.78 + 2.76i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.01 - 3.80i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (11.9 + 3.20i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-5.55 + 5.55i)T - 73iT^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 + (-3.80 - 3.80i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.26 - 12.1i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-0.756 + 2.82i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−10.45656688066698514864535064350, −9.569245381691309197863422731603, −8.642164055012475790366421582211, −7.49292494572539482714820473910, −6.64883329128780646093936167059, −6.12160386116996528522317659652, −5.43579225115932041696372565080, −3.79683548869927700230317356616, −2.84806199754717262964852097656, −1.43659642317059652817097428548,
1.27093107659136422888329259850, 2.48062933035890020130444410197, 3.59299851738028973187056303908, 4.48317452551266482678957031308, 5.91678658316419967017086984926, 6.65824326608576861763332189055, 7.33482806978217797279344870496, 8.798230746171104248726947822637, 9.280900798919088724694529917265, 10.13145839108666175532728815955