L(s) = 1 | + 2-s + 2·3-s + 4-s + 5-s + 2·6-s + 7-s + 8-s + 9-s + 10-s − 11-s + 2·12-s + 6.74·13-s + 14-s + 2·15-s + 16-s − 6.74·17-s + 18-s − 6.74·19-s + 20-s + 2·21-s − 22-s − 6.74·23-s + 2·24-s + 25-s + 6.74·26-s − 4·27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 0.5·4-s + 0.447·5-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.301·11-s + 0.577·12-s + 1.87·13-s + 0.267·14-s + 0.516·15-s + 0.250·16-s − 1.63·17-s + 0.235·18-s − 1.54·19-s + 0.223·20-s + 0.436·21-s − 0.213·22-s − 1.40·23-s + 0.408·24-s + 0.200·25-s + 1.32·26-s − 0.769·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.615293320\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.615293320\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - 2T + 3T^{2} \) |
| 13 | \( 1 - 6.74T + 13T^{2} \) |
| 17 | \( 1 + 6.74T + 17T^{2} \) |
| 19 | \( 1 + 6.74T + 19T^{2} \) |
| 23 | \( 1 + 6.74T + 23T^{2} \) |
| 29 | \( 1 - 8.74T + 29T^{2} \) |
| 31 | \( 1 - 4.74T + 31T^{2} \) |
| 37 | \( 1 - 0.744T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 4.74T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 - 1.25T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 6.74T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−10.55348405566603840281525483099, −9.259740009163592005687544437399, −8.422076705731661692957613300515, −8.094738857572422564011297863893, −6.50910631703067432125052856275, −6.11423148148720500758859467564, −4.61521477311126278475137722645, −3.87372146782149442686012785408, −2.67990546288820200222337348371, −1.83323675261394832793354842974,
1.83323675261394832793354842974, 2.67990546288820200222337348371, 3.87372146782149442686012785408, 4.61521477311126278475137722645, 6.11423148148720500758859467564, 6.50910631703067432125052856275, 8.094738857572422564011297863893, 8.422076705731661692957613300515, 9.259740009163592005687544437399, 10.55348405566603840281525483099