| L(s) = 1 | − 0.438·2-s − 3.68·3-s − 7.80·4-s − 17.8·5-s + 1.61·6-s + 5.43·7-s + 6.93·8-s − 13.4·9-s + 7.80·10-s − 22.4·11-s + 28.7·12-s − 2.38·14-s + 65.6·15-s + 59.4·16-s + 67.9·17-s + 5.88·18-s − 80.8·19-s + 139.·20-s − 20.0·21-s + 9.83·22-s + 140.·23-s − 25.5·24-s + 192.·25-s + 148.·27-s − 42.4·28-s − 106.·29-s − 28.7·30-s + ⋯ |
| L(s) = 1 | − 0.155·2-s − 0.709·3-s − 0.975·4-s − 1.59·5-s + 0.109·6-s + 0.293·7-s + 0.306·8-s − 0.497·9-s + 0.246·10-s − 0.614·11-s + 0.692·12-s − 0.0455·14-s + 1.12·15-s + 0.928·16-s + 0.969·17-s + 0.0770·18-s − 0.975·19-s + 1.55·20-s − 0.208·21-s + 0.0952·22-s + 1.27·23-s − 0.217·24-s + 1.53·25-s + 1.06·27-s − 0.286·28-s − 0.683·29-s − 0.175·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4083583195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4083583195\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 + 0.438T + 8T^{2} \) |
| 3 | \( 1 + 3.68T + 27T^{2} \) |
| 5 | \( 1 + 17.8T + 125T^{2} \) |
| 7 | \( 1 - 5.43T + 343T^{2} \) |
| 11 | \( 1 + 22.4T + 1.33e3T^{2} \) |
| 17 | \( 1 - 67.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 80.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 140.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 106.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 276.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 4.29T + 5.06e4T^{2} \) |
| 41 | \( 1 - 227.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 27.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 318.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 67.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 291.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 663.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 425.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 152.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 117.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 202.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 336.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 718.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 759.T + 9.12e5T^{2} \) |
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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
−12.31346396003444792250656803040, −11.23690681021098228295657930132, −10.64140135859694269461150515758, −9.083339803480987958670912091071, −8.175268189777078009662419863585, −7.36002059665052493565562529895, −5.59169424184138428704291080682, −4.63404211041305517575855897543, −3.44970226597359481152863648505, −0.52691807021939541221250911817,
0.52691807021939541221250911817, 3.44970226597359481152863648505, 4.63404211041305517575855897543, 5.59169424184138428704291080682, 7.36002059665052493565562529895, 8.175268189777078009662419863585, 9.083339803480987958670912091071, 10.64140135859694269461150515758, 11.23690681021098228295657930132, 12.31346396003444792250656803040
