| L(s) = 1 | − 2-s − 1.91·3-s + 4-s + 0.374·5-s + 1.91·6-s − 7-s − 8-s + 0.678·9-s − 0.374·10-s − 1.13·11-s − 1.91·12-s + 0.0849·13-s + 14-s − 0.718·15-s + 16-s − 4.64·17-s − 0.678·18-s + 4.04·19-s + 0.374·20-s + 1.91·21-s + 1.13·22-s + 0.619·23-s + 1.91·24-s − 4.85·25-s − 0.0849·26-s + 4.45·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.10·3-s + 0.5·4-s + 0.167·5-s + 0.782·6-s − 0.377·7-s − 0.353·8-s + 0.226·9-s − 0.118·10-s − 0.341·11-s − 0.553·12-s + 0.0235·13-s + 0.267·14-s − 0.185·15-s + 0.250·16-s − 1.12·17-s − 0.159·18-s + 0.928·19-s + 0.0837·20-s + 0.418·21-s + 0.241·22-s + 0.129·23-s + 0.391·24-s − 0.971·25-s − 0.0166·26-s + 0.856·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4674987109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4674987109\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + 1.91T + 3T^{2} \) |
| 5 | \( 1 - 0.374T + 5T^{2} \) |
| 11 | \( 1 + 1.13T + 11T^{2} \) |
| 13 | \( 1 - 0.0849T + 13T^{2} \) |
| 17 | \( 1 + 4.64T + 17T^{2} \) |
| 19 | \( 1 - 4.04T + 19T^{2} \) |
| 23 | \( 1 - 0.619T + 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 + 2.63T + 31T^{2} \) |
| 37 | \( 1 + 5.53T + 37T^{2} \) |
| 41 | \( 1 - 0.897T + 41T^{2} \) |
| 43 | \( 1 + 3.73T + 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 + 9.09T + 53T^{2} \) |
| 59 | \( 1 - 1.43T + 59T^{2} \) |
| 61 | \( 1 - 7.69T + 61T^{2} \) |
| 67 | \( 1 + 4.04T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 0.0269T + 73T^{2} \) |
| 79 | \( 1 + 3.39T + 79T^{2} \) |
| 83 | \( 1 - 0.337T + 83T^{2} \) |
| 89 | \( 1 + 3.97T + 89T^{2} \) |
| 97 | \( 1 + 5.17T + 97T^{2} \) |
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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
−8.165372150599304032205147800936, −7.24746726176004014042144383755, −6.57734011078434308311801210619, −6.15279270935845511815531961683, −5.27507098411579282258403751235, −4.74405389287157467194864640768, −3.52670913270928080971540301140, −2.65282084056799748604045745364, −1.59058536422875320171964485840, −0.42354721186780244481109080566,
0.42354721186780244481109080566, 1.59058536422875320171964485840, 2.65282084056799748604045745364, 3.52670913270928080971540301140, 4.74405389287157467194864640768, 5.27507098411579282258403751235, 6.15279270935845511815531961683, 6.57734011078434308311801210619, 7.24746726176004014042144383755, 8.165372150599304032205147800936
