| L(s) = 1 | + 1.24·2-s + 3-s − 0.460·4-s − 3.76·5-s + 1.24·6-s − 4.20·7-s − 3.05·8-s + 9-s − 4.66·10-s − 4.74·11-s − 0.460·12-s − 2.41·13-s − 5.22·14-s − 3.76·15-s − 2.86·16-s − 17-s + 1.24·18-s + 2.51·19-s + 1.73·20-s − 4.20·21-s − 5.88·22-s + 1.32·23-s − 3.05·24-s + 9.14·25-s − 3.00·26-s + 27-s + 1.93·28-s + ⋯ |
| L(s) = 1 | + 0.877·2-s + 0.577·3-s − 0.230·4-s − 1.68·5-s + 0.506·6-s − 1.59·7-s − 1.07·8-s + 0.333·9-s − 1.47·10-s − 1.42·11-s − 0.132·12-s − 0.671·13-s − 1.39·14-s − 0.970·15-s − 0.716·16-s − 0.242·17-s + 0.292·18-s + 0.576·19-s + 0.387·20-s − 0.918·21-s − 1.25·22-s + 0.276·23-s − 0.623·24-s + 1.82·25-s − 0.588·26-s + 0.192·27-s + 0.366·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5023829480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5023829480\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 - 1.24T + 2T^{2} \) |
| 5 | \( 1 + 3.76T + 5T^{2} \) |
| 7 | \( 1 + 4.20T + 7T^{2} \) |
| 11 | \( 1 + 4.74T + 11T^{2} \) |
| 13 | \( 1 + 2.41T + 13T^{2} \) |
| 19 | \( 1 - 2.51T + 19T^{2} \) |
| 23 | \( 1 - 1.32T + 23T^{2} \) |
| 29 | \( 1 - 2.31T + 29T^{2} \) |
| 31 | \( 1 + 3.84T + 31T^{2} \) |
| 37 | \( 1 + 4.08T + 37T^{2} \) |
| 41 | \( 1 + 3.53T + 41T^{2} \) |
| 43 | \( 1 + 0.802T + 43T^{2} \) |
| 47 | \( 1 + 2.93T + 47T^{2} \) |
| 53 | \( 1 - 4.31T + 53T^{2} \) |
| 59 | \( 1 - 8.38T + 59T^{2} \) |
| 61 | \( 1 + 6.40T + 61T^{2} \) |
| 67 | \( 1 + 3.80T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 83 | \( 1 + 0.0914T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 5.83T + 97T^{2} \) |
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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
−8.509952647063671542257167120070, −7.48819956288405664118181373143, −7.19958018507401521062132572733, −6.18206338250730594029490671731, −5.19710328200262187390531521414, −4.56748170532729210131978968698, −3.69090434925946719205655709397, −3.19763245632341357079765663994, −2.65619838960272581386174186698, −0.31929918945365498160767165766,
0.31929918945365498160767165766, 2.65619838960272581386174186698, 3.19763245632341357079765663994, 3.69090434925946719205655709397, 4.56748170532729210131978968698, 5.19710328200262187390531521414, 6.18206338250730594029490671731, 7.19958018507401521062132572733, 7.48819956288405664118181373143, 8.509952647063671542257167120070
