| L(s) = 1 | + 2-s + 2.29·3-s + 4-s − 0.901·5-s + 2.29·6-s + 7-s + 8-s + 2.24·9-s − 0.901·10-s + 4.33·11-s + 2.29·12-s + 14-s − 2.06·15-s + 16-s + 5.06·17-s + 2.24·18-s − 6.17·19-s − 0.901·20-s + 2.29·21-s + 4.33·22-s + 8.45·23-s + 2.29·24-s − 4.18·25-s − 1.72·27-s + 28-s − 2.19·29-s − 2.06·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.32·3-s + 0.5·4-s − 0.403·5-s + 0.935·6-s + 0.377·7-s + 0.353·8-s + 0.749·9-s − 0.285·10-s + 1.30·11-s + 0.661·12-s + 0.267·14-s − 0.533·15-s + 0.250·16-s + 1.22·17-s + 0.529·18-s − 1.41·19-s − 0.201·20-s + 0.499·21-s + 0.924·22-s + 1.76·23-s + 0.467·24-s − 0.837·25-s − 0.331·27-s + 0.188·28-s − 0.407·29-s − 0.377·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.712153033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.712153033\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.29T + 3T^{2} \) |
| 5 | \( 1 + 0.901T + 5T^{2} \) |
| 11 | \( 1 - 4.33T + 11T^{2} \) |
| 17 | \( 1 - 5.06T + 17T^{2} \) |
| 19 | \( 1 + 6.17T + 19T^{2} \) |
| 23 | \( 1 - 8.45T + 23T^{2} \) |
| 29 | \( 1 + 2.19T + 29T^{2} \) |
| 31 | \( 1 + 0.873T + 31T^{2} \) |
| 37 | \( 1 + 0.144T + 37T^{2} \) |
| 41 | \( 1 + 3.99T + 41T^{2} \) |
| 43 | \( 1 - 7.70T + 43T^{2} \) |
| 47 | \( 1 + 2.92T + 47T^{2} \) |
| 53 | \( 1 - 1.69T + 53T^{2} \) |
| 59 | \( 1 + 8.54T + 59T^{2} \) |
| 61 | \( 1 - 8.33T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 3.27T + 71T^{2} \) |
| 73 | \( 1 - 0.539T + 73T^{2} \) |
| 79 | \( 1 - 6.53T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 7.79T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.945612839210287355035305080330, −8.156075940329928205694524236149, −7.53001876927068264953707765647, −6.75234725185590035579240500843, −5.83373445020261635972952808985, −4.76300378564729791701014357923, −3.88037780427279370265719252982, −3.41620986211026416325546705310, −2.38072164457690162941487666136, −1.37055758538464057392404091887,
1.37055758538464057392404091887, 2.38072164457690162941487666136, 3.41620986211026416325546705310, 3.88037780427279370265719252982, 4.76300378564729791701014357923, 5.83373445020261635972952808985, 6.75234725185590035579240500843, 7.53001876927068264953707765647, 8.156075940329928205694524236149, 8.945612839210287355035305080330
