| L(s) = 1 | − 3.83·2-s − 0.279·3-s + 6.70·4-s + 11.3·5-s + 1.07·6-s + 31.0·7-s + 4.97·8-s − 26.9·9-s − 43.6·10-s − 20.9·11-s − 1.87·12-s − 119.·14-s − 3.17·15-s − 72.6·16-s + 114.·17-s + 103.·18-s − 45.1·19-s + 76.2·20-s − 8.67·21-s + 80.3·22-s + 73.9·23-s − 1.38·24-s + 4.29·25-s + 15.0·27-s + 208.·28-s − 27.2·29-s + 12.1·30-s + ⋯ |
| L(s) = 1 | − 1.35·2-s − 0.0537·3-s + 0.837·4-s + 1.01·5-s + 0.0728·6-s + 1.67·7-s + 0.219·8-s − 0.997·9-s − 1.37·10-s − 0.574·11-s − 0.0450·12-s − 2.27·14-s − 0.0546·15-s − 1.13·16-s + 1.63·17-s + 1.35·18-s − 0.545·19-s + 0.852·20-s − 0.0901·21-s + 0.778·22-s + 0.670·23-s − 0.0118·24-s + 0.0343·25-s + 0.107·27-s + 1.40·28-s − 0.174·29-s + 0.0740·30-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
| \(L(2)\) |
\(\approx\) |
\(1.102154135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.102154135\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 3.83T + 8T^{2} \) |
| 3 | \( 1 + 0.279T + 27T^{2} \) |
| 5 | \( 1 - 11.3T + 125T^{2} \) |
| 7 | \( 1 - 31.0T + 343T^{2} \) |
| 11 | \( 1 + 20.9T + 1.33e3T^{2} \) |
| 17 | \( 1 - 114.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 45.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 73.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 27.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 179.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 354.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 81.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 256.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 463.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 76.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 54.4T + 2.05e5T^{2} \) |
| 61 | \( 1 + 494.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 611.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 16.3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 321.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 385.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 663.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 545.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 689.T + 9.12e5T^{2} \) |
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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
−11.91615836260454359289041917430, −10.97547178961279448559540462072, −10.25937622091534888311899548558, −9.214598970884098471984993317451, −8.246407728181754191017799628262, −7.65098254202619558744935977826, −5.91301392371351066516395141974, −4.88839461256788786230164986504, −2.38805935430924639911246460856, −1.10824462492910475164191104661,
1.10824462492910475164191104661, 2.38805935430924639911246460856, 4.88839461256788786230164986504, 5.91301392371351066516395141974, 7.65098254202619558744935977826, 8.246407728181754191017799628262, 9.214598970884098471984993317451, 10.25937622091534888311899548558, 10.97547178961279448559540462072, 11.91615836260454359289041917430
