| L(s) = 1 | + 2-s − 1.51·3-s + 4-s − 1.33·5-s − 1.51·6-s − 3.41·7-s + 8-s − 0.705·9-s − 1.33·10-s − 1.69·11-s − 1.51·12-s − 1.37·13-s − 3.41·14-s + 2.01·15-s + 16-s + 3.29·17-s − 0.705·18-s − 19-s − 1.33·20-s + 5.16·21-s − 1.69·22-s − 9.07·23-s − 1.51·24-s − 3.22·25-s − 1.37·26-s + 5.61·27-s − 3.41·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.874·3-s + 0.5·4-s − 0.594·5-s − 0.618·6-s − 1.28·7-s + 0.353·8-s − 0.235·9-s − 0.420·10-s − 0.512·11-s − 0.437·12-s − 0.381·13-s − 0.912·14-s + 0.520·15-s + 0.250·16-s + 0.799·17-s − 0.166·18-s − 0.229·19-s − 0.297·20-s + 1.12·21-s − 0.362·22-s − 1.89·23-s − 0.309·24-s − 0.645·25-s − 0.269·26-s + 1.08·27-s − 0.644·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4179489258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4179489258\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 1.51T + 3T^{2} \) |
| 5 | \( 1 + 1.33T + 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 + 1.69T + 11T^{2} \) |
| 13 | \( 1 + 1.37T + 13T^{2} \) |
| 17 | \( 1 - 3.29T + 17T^{2} \) |
| 23 | \( 1 + 9.07T + 23T^{2} \) |
| 29 | \( 1 + 4.00T + 29T^{2} \) |
| 31 | \( 1 + 3.30T + 31T^{2} \) |
| 37 | \( 1 + 7.04T + 37T^{2} \) |
| 41 | \( 1 - 3.82T + 41T^{2} \) |
| 43 | \( 1 + 2.87T + 43T^{2} \) |
| 47 | \( 1 - 0.403T + 47T^{2} \) |
| 53 | \( 1 + 3.47T + 53T^{2} \) |
| 59 | \( 1 + 4.97T + 59T^{2} \) |
| 61 | \( 1 - 5.81T + 61T^{2} \) |
| 67 | \( 1 - 2.62T + 67T^{2} \) |
| 71 | \( 1 + 1.72T + 71T^{2} \) |
| 73 | \( 1 + 2.94T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 0.930T + 89T^{2} \) |
| 97 | \( 1 - 19.0T + 97T^{2} \) |
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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
−7.62772273274959311684048477253, −7.01934461528663616455434517000, −6.17243312278070017756997352913, −5.81251456232009463192142848344, −5.20004441363032903817556136081, −4.24260411907468741992914528078, −3.58655297990964335806155694932, −2.93599555216577329901167928122, −1.90226712690623792675605852810, −0.28006036995578447895045119442,
0.28006036995578447895045119442, 1.90226712690623792675605852810, 2.93599555216577329901167928122, 3.58655297990964335806155694932, 4.24260411907468741992914528078, 5.20004441363032903817556136081, 5.81251456232009463192142848344, 6.17243312278070017756997352913, 7.01934461528663616455434517000, 7.62772273274959311684048477253
