| L(s) = 1 | + 4.67·2-s + 7.62·3-s + 13.8·4-s + 11.9·5-s + 35.6·6-s − 26.1·7-s + 27.1·8-s + 31.2·9-s + 55.6·10-s + 3.24·11-s + 105.·12-s − 20.0·13-s − 122.·14-s + 90.9·15-s + 16.4·16-s + 145.·18-s + 57.3·19-s + 164.·20-s − 199.·21-s + 15.1·22-s − 77.0·23-s + 207.·24-s + 17.0·25-s − 93.6·26-s + 32.1·27-s − 361.·28-s + 286.·29-s + ⋯ |
| L(s) = 1 | + 1.65·2-s + 1.46·3-s + 1.72·4-s + 1.06·5-s + 2.42·6-s − 1.41·7-s + 1.20·8-s + 1.15·9-s + 1.76·10-s + 0.0889·11-s + 2.53·12-s − 0.427·13-s − 2.32·14-s + 1.56·15-s + 0.257·16-s + 1.90·18-s + 0.692·19-s + 1.84·20-s − 2.07·21-s + 0.146·22-s − 0.698·23-s + 1.76·24-s + 0.136·25-s − 0.706·26-s + 0.229·27-s − 2.43·28-s + 1.83·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.326407266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.326407266\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 4.67T + 8T^{2} \) |
| 3 | \( 1 - 7.62T + 27T^{2} \) |
| 5 | \( 1 - 11.9T + 125T^{2} \) |
| 7 | \( 1 + 26.1T + 343T^{2} \) |
| 11 | \( 1 - 3.24T + 1.33e3T^{2} \) |
| 13 | \( 1 + 20.0T + 2.19e3T^{2} \) |
| 19 | \( 1 - 57.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 77.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 286.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 8.54T + 2.97e4T^{2} \) |
| 37 | \( 1 + 357.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 194.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 74.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 23.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 104.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 249.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 370.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 939.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 520.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 348.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 953.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.41e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 486.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 685.T + 9.12e5T^{2} \) |
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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
−11.91226924106943251902957625693, −10.16152795725608277697571542451, −9.645152115890941962420273654958, −8.558900681116993448199272783299, −7.06120151071962756761984013898, −6.29819334456106865558532932530, −5.20679053668716813955100439581, −3.76790985156087331662988422445, −2.99353576388426351759362661057, −2.14031467337239665624824831578,
2.14031467337239665624824831578, 2.99353576388426351759362661057, 3.76790985156087331662988422445, 5.20679053668716813955100439581, 6.29819334456106865558532932530, 7.06120151071962756761984013898, 8.558900681116993448199272783299, 9.645152115890941962420273654958, 10.16152795725608277697571542451, 11.91226924106943251902957625693
