L(s) = 1 | + 9·3-s − 96.3·5-s + 146.·7-s + 81·9-s + 121·11-s − 710.·13-s − 867.·15-s + 526.·17-s − 2.31e3·19-s + 1.32e3·21-s + 2.15e3·23-s + 6.16e3·25-s + 729·27-s − 2.94e3·29-s + 4.09e3·31-s + 1.08e3·33-s − 1.41e4·35-s + 7.88e3·37-s − 6.39e3·39-s − 7.19e3·41-s − 2.23e4·43-s − 7.80e3·45-s + 998.·47-s + 4.76e3·49-s + 4.73e3·51-s + 3.99e4·53-s − 1.16e4·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.72·5-s + 1.13·7-s + 0.333·9-s + 0.301·11-s − 1.16·13-s − 0.995·15-s + 0.441·17-s − 1.46·19-s + 0.654·21-s + 0.849·23-s + 1.97·25-s + 0.192·27-s − 0.649·29-s + 0.766·31-s + 0.174·33-s − 1.95·35-s + 0.946·37-s − 0.673·39-s − 0.668·41-s − 1.84·43-s − 0.574·45-s + 0.0659·47-s + 0.283·49-s + 0.255·51-s + 1.95·53-s − 0.519·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.837475475\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.837475475\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 11 | \( 1 - 121T \) |
good | 5 | \( 1 + 96.3T + 3.12e3T^{2} \) |
| 7 | \( 1 - 146.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 710.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 526.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.31e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.15e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.94e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.09e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.88e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.19e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.23e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 998.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.99e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.17e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.20e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.84e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 8.35e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.05e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.16e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.74e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.90e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.78e5T + 8.58e9T^{2} \) |
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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
−10.12769017643456809478009207364, −8.848195857139494911584127682091, −8.194421177483530503264083317200, −7.59369537165670910311533815191, −6.77544256230302727003592579482, −4.97889621244425813129816946195, −4.35615476742500602791470476734, −3.38104243628018428934261223314, −2.10689845048692169427556051117, −0.65181924634832009121885718511,
0.65181924634832009121885718511, 2.10689845048692169427556051117, 3.38104243628018428934261223314, 4.35615476742500602791470476734, 4.97889621244425813129816946195, 6.77544256230302727003592579482, 7.59369537165670910311533815191, 8.194421177483530503264083317200, 8.848195857139494911584127682091, 10.12769017643456809478009207364