| L(s) = 1 | − 2·2-s + 7.15·3-s + 4·4-s − 8.30·5-s − 14.3·6-s + 35.1·7-s − 8·8-s + 24.1·9-s + 16.6·10-s + 18.3·11-s + 28.6·12-s − 40.0·13-s − 70.3·14-s − 59.3·15-s + 16·16-s − 125.·17-s − 48.3·18-s − 19·19-s − 33.2·20-s + 251.·21-s − 36.6·22-s + 8.97·23-s − 57.2·24-s − 56.0·25-s + 80.1·26-s − 20.3·27-s + 140.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.37·3-s + 0.5·4-s − 0.742·5-s − 0.973·6-s + 1.89·7-s − 0.353·8-s + 0.894·9-s + 0.525·10-s + 0.501·11-s + 0.688·12-s − 0.854·13-s − 1.34·14-s − 1.02·15-s + 0.250·16-s − 1.78·17-s − 0.632·18-s − 0.229·19-s − 0.371·20-s + 2.61·21-s − 0.354·22-s + 0.0813·23-s − 0.486·24-s − 0.448·25-s + 0.604·26-s − 0.145·27-s + 0.949·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.379965944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.379965944\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 - 7.15T + 27T^{2} \) |
| 5 | \( 1 + 8.30T + 125T^{2} \) |
| 7 | \( 1 - 35.1T + 343T^{2} \) |
| 11 | \( 1 - 18.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 40.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 125.T + 4.91e3T^{2} \) |
| 23 | \( 1 - 8.97T + 1.21e4T^{2} \) |
| 29 | \( 1 - 153.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 114.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 83.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 355.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 467.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 166.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 258.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 371.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 47.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 755.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 349.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 54.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 438.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.07e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 501.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.43e3T + 9.12e5T^{2} \) |
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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
−15.44249293269138385164643740783, −14.83341150220674979248969608990, −13.84081462250573773073310298309, −11.93686823526188039545269815241, −10.89036109132546340838310197267, −9.072963783034075895055675876009, −8.266695078387097207180441493137, −7.35742288151623467179702724045, −4.37425361656522483900048009165, −2.14164103742936405307762083682,
2.14164103742936405307762083682, 4.37425361656522483900048009165, 7.35742288151623467179702724045, 8.266695078387097207180441493137, 9.072963783034075895055675876009, 10.89036109132546340838310197267, 11.93686823526188039545269815241, 13.84081462250573773073310298309, 14.83341150220674979248969608990, 15.44249293269138385164643740783
