| L(s) = 1 | + 1.81·2-s + 3·3-s − 4.69·4-s − 19.2·5-s + 5.45·6-s + 14.1·7-s − 23.0·8-s + 9·9-s − 34.9·10-s − 14.0·12-s + 54.2·13-s + 25.6·14-s − 57.6·15-s − 4.42·16-s + 78.2·17-s + 16.3·18-s + 90.2·19-s + 90.1·20-s + 42.3·21-s − 122.·23-s − 69.2·24-s + 244.·25-s + 98.6·26-s + 27·27-s − 66.1·28-s − 123.·29-s − 104.·30-s + ⋯ |
| L(s) = 1 | + 0.642·2-s + 0.577·3-s − 0.586·4-s − 1.71·5-s + 0.371·6-s + 0.761·7-s − 1.02·8-s + 0.333·9-s − 1.10·10-s − 0.338·12-s + 1.15·13-s + 0.489·14-s − 0.992·15-s − 0.0691·16-s + 1.11·17-s + 0.214·18-s + 1.08·19-s + 1.00·20-s + 0.439·21-s − 1.11·23-s − 0.588·24-s + 1.95·25-s + 0.744·26-s + 0.192·27-s − 0.446·28-s − 0.792·29-s − 0.638·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.147366727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.147366727\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.81T + 8T^{2} \) |
| 5 | \( 1 + 19.2T + 125T^{2} \) |
| 7 | \( 1 - 14.1T + 343T^{2} \) |
| 13 | \( 1 - 54.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 78.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 90.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 123.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 212.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 429.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 224.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 408.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 224.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 283.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 306.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 111.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 153.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 323.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 194.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 788.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 870.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 788.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 852.T + 9.12e5T^{2} \) |
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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
−11.46240094858005845463129085628, −10.06153550009571196134190009322, −8.889356672835282737270399701201, −8.040018462962733461985957115522, −7.64874480450264325862200703321, −5.96948621316237223497226329494, −4.70253144711563399570300509452, −3.90997729529186602117119618084, −3.18336058765189940281318337804, −0.910446496688850171119099671549,
0.910446496688850171119099671549, 3.18336058765189940281318337804, 3.90997729529186602117119618084, 4.70253144711563399570300509452, 5.96948621316237223497226329494, 7.64874480450264325862200703321, 8.040018462962733461985957115522, 8.889356672835282737270399701201, 10.06153550009571196134190009322, 11.46240094858005845463129085628
