| L(s) = 1 | + (−1.00 − 1.73i)2-s + (−2.96 + 0.463i)3-s + (−1.98 + 3.47i)4-s + (0.608 + 3.44i)5-s + (3.77 + 4.66i)6-s + (5.71 − 6.81i)7-s + (7.99 − 0.0425i)8-s + (8.56 − 2.74i)9-s + (5.35 − 4.51i)10-s + (−5.38 − 0.949i)11-s + (4.28 − 11.2i)12-s + (21.3 + 7.76i)13-s + (−17.5 − 3.05i)14-s + (−3.40 − 9.93i)15-s + (−8.09 − 13.7i)16-s + (−8.12 − 14.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.501 − 0.865i)2-s + (−0.987 + 0.154i)3-s + (−0.496 + 0.867i)4-s + (0.121 + 0.689i)5-s + (0.629 + 0.777i)6-s + (0.816 − 0.973i)7-s + (0.999 − 0.00532i)8-s + (0.952 − 0.305i)9-s + (0.535 − 0.451i)10-s + (−0.489 − 0.0862i)11-s + (0.356 − 0.934i)12-s + (1.64 + 0.597i)13-s + (−1.25 − 0.218i)14-s + (−0.226 − 0.662i)15-s + (−0.506 − 0.862i)16-s + (−0.478 − 0.827i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 + 0.594i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.804 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.821448 - 0.270501i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.821448 - 0.270501i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.00 + 1.73i)T \) |
| 3 | \( 1 + (2.96 - 0.463i)T \) |
| good | 5 | \( 1 + (-0.608 - 3.44i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-5.71 + 6.81i)T + (-8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (5.38 + 0.949i)T + (113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (-21.3 - 7.76i)T + (129. + 108. i)T^{2} \) |
| 17 | \( 1 + (8.12 + 14.0i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-19.5 - 11.2i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-22.1 - 26.4i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-24.1 + 8.78i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (14.3 + 17.1i)T + (-166. + 946. i)T^{2} \) |
| 37 | \( 1 + (-7.88 - 13.6i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (49.6 + 18.0i)T + (1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (9.61 + 1.69i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (14.9 - 17.8i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 4.09T + 2.80e3T^{2} \) |
| 59 | \( 1 + (28.0 - 4.93i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (31.1 + 26.1i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-30.9 + 84.9i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-87.8 + 50.7i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (16.0 - 27.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-8.65 - 23.7i)T + (-4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-16.3 - 45.0i)T + (-5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-45.0 + 78.0i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (25.7 - 146. i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−13.35561884409425903815276469738, −11.79665126139749831907216981468, −11.08816062029150008538572434241, −10.60067300448589859155195208797, −9.411453784694933021041178848428, −7.84535903001109513116262543051, −6.75919718431047098432956506802, −4.95707813850934156373394867789, −3.59013581963155481392979302190, −1.23219342697925501359653085036,
1.22484520985716295697149610183, 4.85518941160000261011748123199, 5.54004615391942741589164336343, 6.69570652097413241042284358377, 8.246499916303068544056018055478, 8.884629742353141177252222021529, 10.47834822115065429089987760379, 11.26156766248850951563222544790, 12.65903537943598917465308794118, 13.45178528726380736555815172843
