| L(s) = 1 | + 0.397·2-s − 1.55·3-s − 1.84·4-s + 0.426·5-s − 0.616·6-s − 5.04·7-s − 1.52·8-s − 0.592·9-s + 0.169·10-s − 1.86·11-s + 2.85·12-s − 13-s − 2.00·14-s − 0.661·15-s + 3.07·16-s − 1.84·17-s − 0.235·18-s − 2.20·19-s − 0.785·20-s + 7.82·21-s − 0.740·22-s − 0.519·23-s + 2.36·24-s − 4.81·25-s − 0.397·26-s + 5.57·27-s + 9.29·28-s + ⋯ |
| L(s) = 1 | + 0.280·2-s − 0.895·3-s − 0.921·4-s + 0.190·5-s − 0.251·6-s − 1.90·7-s − 0.539·8-s − 0.197·9-s + 0.0535·10-s − 0.562·11-s + 0.825·12-s − 0.277·13-s − 0.535·14-s − 0.170·15-s + 0.769·16-s − 0.446·17-s − 0.0554·18-s − 0.505·19-s − 0.175·20-s + 1.70·21-s − 0.157·22-s − 0.108·23-s + 0.483·24-s − 0.963·25-s − 0.0779·26-s + 1.07·27-s + 1.75·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3212037278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3212037278\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 0.397T + 2T^{2} \) |
| 3 | \( 1 + 1.55T + 3T^{2} \) |
| 5 | \( 1 - 0.426T + 5T^{2} \) |
| 7 | \( 1 + 5.04T + 7T^{2} \) |
| 11 | \( 1 + 1.86T + 11T^{2} \) |
| 17 | \( 1 + 1.84T + 17T^{2} \) |
| 19 | \( 1 + 2.20T + 19T^{2} \) |
| 23 | \( 1 + 0.519T + 23T^{2} \) |
| 29 | \( 1 - 3.88T + 29T^{2} \) |
| 31 | \( 1 - 1.35T + 31T^{2} \) |
| 37 | \( 1 - 1.20T + 37T^{2} \) |
| 41 | \( 1 - 3.92T + 41T^{2} \) |
| 43 | \( 1 + 6.72T + 43T^{2} \) |
| 47 | \( 1 + 7.79T + 47T^{2} \) |
| 53 | \( 1 + 2.60T + 53T^{2} \) |
| 59 | \( 1 + 4.73T + 59T^{2} \) |
| 61 | \( 1 - 8.93T + 61T^{2} \) |
| 67 | \( 1 - 7.55T + 67T^{2} \) |
| 71 | \( 1 - 7.41T + 71T^{2} \) |
| 73 | \( 1 + 4.39T + 73T^{2} \) |
| 79 | \( 1 - 1.81T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 3.54T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−9.842655355081382059006035714632, −8.952227825016913201309771051821, −8.107665750581967021604981896105, −6.77049952838596715991623272857, −6.19072740253299790116002507640, −5.52876445921158925618375701095, −4.62871245897230949781799327615, −3.58432564430135959061169281567, −2.66685506748308266781587501819, −0.38881518243556519866211407715,
0.38881518243556519866211407715, 2.66685506748308266781587501819, 3.58432564430135959061169281567, 4.62871245897230949781799327615, 5.52876445921158925618375701095, 6.19072740253299790116002507640, 6.77049952838596715991623272857, 8.107665750581967021604981896105, 8.952227825016913201309771051821, 9.842655355081382059006035714632
