| L(s) = 1 | − 4.59·2-s − 3·3-s + 13.1·4-s − 0.882·5-s + 13.7·6-s − 25.5·7-s − 23.5·8-s + 9·9-s + 4.05·10-s + 9.51·11-s − 39.3·12-s + 17.9·13-s + 117.·14-s + 2.64·15-s + 3.11·16-s − 75.2·17-s − 41.3·18-s + 153.·19-s − 11.5·20-s + 76.6·21-s − 43.7·22-s − 112.·23-s + 70.5·24-s − 124.·25-s − 82.4·26-s − 27·27-s − 335.·28-s + ⋯ |
| L(s) = 1 | − 1.62·2-s − 0.577·3-s + 1.63·4-s − 0.0789·5-s + 0.938·6-s − 1.37·7-s − 1.03·8-s + 0.333·9-s + 0.128·10-s + 0.260·11-s − 0.946·12-s + 0.382·13-s + 2.24·14-s + 0.0455·15-s + 0.0487·16-s − 1.07·17-s − 0.541·18-s + 1.85·19-s − 0.129·20-s + 0.796·21-s − 0.423·22-s − 1.01·23-s + 0.599·24-s − 0.993·25-s − 0.621·26-s − 0.192·27-s − 2.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1011 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3094383625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3094383625\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 337 | \( 1 + 337T \) |
| good | 2 | \( 1 + 4.59T + 8T^{2} \) |
| 5 | \( 1 + 0.882T + 125T^{2} \) |
| 7 | \( 1 + 25.5T + 343T^{2} \) |
| 11 | \( 1 - 9.51T + 1.33e3T^{2} \) |
| 13 | \( 1 - 17.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 75.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 153.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 112.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 194.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 190.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 391.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 399.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 384.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 253.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 30.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 367.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 394.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 629.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 483.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 486.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 303.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 426.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 439.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.23e3T + 9.12e5T^{2} \) |
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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
−9.439896324288707238545846035100, −9.110020675192341189206494487504, −7.939347987037670847135877772698, −7.16896614886979610233874425887, −6.46439556041352306583555803369, −5.68663077828749649171488812569, −4.14499575610957732278936071442, −2.94766580347995575173623045015, −1.58952475936999339652504145391, −0.39484163544998363462169014139,
0.39484163544998363462169014139, 1.58952475936999339652504145391, 2.94766580347995575173623045015, 4.14499575610957732278936071442, 5.68663077828749649171488812569, 6.46439556041352306583555803369, 7.16896614886979610233874425887, 7.939347987037670847135877772698, 9.110020675192341189206494487504, 9.439896324288707238545846035100
