| L(s) = 1 | − 2.16·2-s + 9.14·3-s − 3.31·4-s + 16.2·5-s − 19.7·6-s + 12.2·7-s + 24.4·8-s + 56.5·9-s − 35.1·10-s − 11.1·11-s − 30.2·12-s + 28.8·13-s − 26.4·14-s + 148.·15-s − 26.5·16-s − 122.·18-s − 79.5·19-s − 53.7·20-s + 111.·21-s + 24.1·22-s − 45.0·23-s + 223.·24-s + 138.·25-s − 62.5·26-s + 270.·27-s − 40.5·28-s + 20.3·29-s + ⋯ |
| L(s) = 1 | − 0.765·2-s + 1.75·3-s − 0.414·4-s + 1.45·5-s − 1.34·6-s + 0.660·7-s + 1.08·8-s + 2.09·9-s − 1.11·10-s − 0.306·11-s − 0.728·12-s + 0.616·13-s − 0.505·14-s + 2.55·15-s − 0.414·16-s − 1.60·18-s − 0.960·19-s − 0.601·20-s + 1.16·21-s + 0.234·22-s − 0.408·23-s + 1.90·24-s + 1.10·25-s − 0.471·26-s + 1.92·27-s − 0.273·28-s + 0.130·29-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
| \(L(2)\) |
\(\approx\) |
\(2.783017591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.783017591\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 2.16T + 8T^{2} \) |
| 3 | \( 1 - 9.14T + 27T^{2} \) |
| 5 | \( 1 - 16.2T + 125T^{2} \) |
| 7 | \( 1 - 12.2T + 343T^{2} \) |
| 11 | \( 1 + 11.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 28.8T + 2.19e3T^{2} \) |
| 19 | \( 1 + 79.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 45.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 20.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 1.03T + 2.97e4T^{2} \) |
| 37 | \( 1 + 219.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 310.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 483.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 632.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 490.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 147.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 176.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 809.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 714.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 780.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 230.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 236.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 688.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.84e3T + 9.12e5T^{2} \) |
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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
−10.81217535844956751247945212966, −9.988855277190082740388463912366, −9.388693851775112223697830677177, −8.488434159587181295653221589465, −8.120068912494754485503792789982, −6.77332377360380800406156412236, −5.17636498002881724781946133171, −3.88501664806996189253614872457, −2.29929516347426127831644346678, −1.49923833465031044330113644005,
1.49923833465031044330113644005, 2.29929516347426127831644346678, 3.88501664806996189253614872457, 5.17636498002881724781946133171, 6.77332377360380800406156412236, 8.120068912494754485503792789982, 8.488434159587181295653221589465, 9.388693851775112223697830677177, 9.988855277190082740388463912366, 10.81217535844956751247945212966
