| L(s) = 1 | + 3.25·2-s − 6.51·3-s + 2.61·4-s + 19.1·5-s − 21.2·6-s + 22.0·7-s − 17.5·8-s + 15.4·9-s + 62.2·10-s − 8.25·11-s − 17.0·12-s − 0.398·13-s + 71.9·14-s − 124.·15-s − 78.0·16-s + 50.4·18-s + 103.·19-s + 49.8·20-s − 144.·21-s − 26.9·22-s + 138.·23-s + 114.·24-s + 240.·25-s − 1.29·26-s + 75.0·27-s + 57.7·28-s + 222.·29-s + ⋯ |
| L(s) = 1 | + 1.15·2-s − 1.25·3-s + 0.326·4-s + 1.70·5-s − 1.44·6-s + 1.19·7-s − 0.775·8-s + 0.573·9-s + 1.96·10-s − 0.226·11-s − 0.409·12-s − 0.00849·13-s + 1.37·14-s − 2.14·15-s − 1.21·16-s + 0.660·18-s + 1.24·19-s + 0.557·20-s − 1.49·21-s − 0.260·22-s + 1.25·23-s + 0.973·24-s + 1.92·25-s − 0.00978·26-s + 0.535·27-s + 0.389·28-s + 1.42·29-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
| \(L(2)\) |
\(\approx\) |
\(2.996766811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.996766811\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 3.25T + 8T^{2} \) |
| 3 | \( 1 + 6.51T + 27T^{2} \) |
| 5 | \( 1 - 19.1T + 125T^{2} \) |
| 7 | \( 1 - 22.0T + 343T^{2} \) |
| 11 | \( 1 + 8.25T + 1.33e3T^{2} \) |
| 13 | \( 1 + 0.398T + 2.19e3T^{2} \) |
| 19 | \( 1 - 103.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 138.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 222.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 49.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 61.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 387.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 20.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 44.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 59.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 238.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 595.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 408.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 22.3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 682.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 312.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 904.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.00e3T + 9.12e5T^{2} \) |
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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
−11.55835058102550832438604728566, −10.69866048858937708251406068773, −9.732620335319081631551907324454, −8.602050310712231841769269455271, −6.84892644589831965890139304254, −5.95578479217814313661271314419, −5.15303137453054569643166417598, −4.85409832371727944274140048009, −2.80688838355943324952311643770, −1.22431259220384119119458996674,
1.22431259220384119119458996674, 2.80688838355943324952311643770, 4.85409832371727944274140048009, 5.15303137453054569643166417598, 5.95578479217814313661271314419, 6.84892644589831965890139304254, 8.602050310712231841769269455271, 9.732620335319081631551907324454, 10.69866048858937708251406068773, 11.55835058102550832438604728566
