L(s) = 1 | + 2-s + 2.41·3-s + 4-s + 3.82·5-s + 2.41·6-s + 8-s + 2.82·9-s + 3.82·10-s + 2.24·11-s + 2.41·12-s + 3.41·13-s + 9.24·15-s + 16-s − 1.82·17-s + 2.82·18-s − 2.82·19-s + 3.82·20-s + 2.24·22-s − 5.65·23-s + 2.41·24-s + 9.65·25-s + 3.41·26-s − 0.414·27-s − 5·29-s + 9.24·30-s − 8.41·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.39·3-s + 0.5·4-s + 1.71·5-s + 0.985·6-s + 0.353·8-s + 0.942·9-s + 1.21·10-s + 0.676·11-s + 0.696·12-s + 0.946·13-s + 2.38·15-s + 0.250·16-s − 0.443·17-s + 0.666·18-s − 0.648·19-s + 0.856·20-s + 0.478·22-s − 1.17·23-s + 0.492·24-s + 1.93·25-s + 0.669·26-s − 0.0797·27-s − 0.928·29-s + 1.68·30-s − 1.51·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.026339463\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.026339463\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 - 2.41T + 3T^{2} \) |
| 5 | \( 1 - 3.82T + 5T^{2} \) |
| 11 | \( 1 - 2.24T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 + 1.82T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 8.41T + 31T^{2} \) |
| 37 | \( 1 - 9.41T + 37T^{2} \) |
| 43 | \( 1 + 7.24T + 43T^{2} \) |
| 47 | \( 1 - 1.75T + 47T^{2} \) |
| 53 | \( 1 - 9.48T + 53T^{2} \) |
| 59 | \( 1 + 6.48T + 59T^{2} \) |
| 61 | \( 1 + 5.48T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 0.757T + 79T^{2} \) |
| 83 | \( 1 - 5.75T + 83T^{2} \) |
| 89 | \( 1 + 1.82T + 89T^{2} \) |
| 97 | \( 1 - 5.82T + 97T^{2} \) |
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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
−8.644907930985845582965819274730, −7.72942534799284651813487818808, −6.86123036651845838446535259478, −6.01327976419981801911694296325, −5.74869129574092192796578771888, −4.46304335321314618777407361582, −3.76093908948347490920581515469, −2.92310639294324702810369196449, −1.99748729167342234837029981045, −1.64982416714107167515857975677,
1.64982416714107167515857975677, 1.99748729167342234837029981045, 2.92310639294324702810369196449, 3.76093908948347490920581515469, 4.46304335321314618777407361582, 5.74869129574092192796578771888, 6.01327976419981801911694296325, 6.86123036651845838446535259478, 7.72942534799284651813487818808, 8.644907930985845582965819274730