L(s) = 1 | − 2.21·3-s + 3.59·5-s + 5.06·7-s + 1.90·9-s + 2.55·11-s + 4.93·13-s − 7.96·15-s − 2.68·17-s − 4.72·19-s − 11.2·21-s − 1.49·23-s + 7.93·25-s + 2.41·27-s − 2.43·29-s + 3.19·31-s − 5.67·33-s + 18.2·35-s + 2.90·37-s − 10.9·39-s − 7.62·43-s + 6.86·45-s − 5.15·47-s + 18.6·49-s + 5.95·51-s + 11.1·53-s + 9.20·55-s + 10.4·57-s + ⋯ |
L(s) = 1 | − 1.27·3-s + 1.60·5-s + 1.91·7-s + 0.636·9-s + 0.771·11-s + 1.36·13-s − 2.05·15-s − 0.651·17-s − 1.08·19-s − 2.44·21-s − 0.311·23-s + 1.58·25-s + 0.465·27-s − 0.451·29-s + 0.573·31-s − 0.987·33-s + 3.07·35-s + 0.478·37-s − 1.75·39-s − 1.16·43-s + 1.02·45-s − 0.751·47-s + 2.66·49-s + 0.833·51-s + 1.52·53-s + 1.24·55-s + 1.38·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.604019520\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.604019520\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 41 | \( 1 \) |
good | 3 | \( 1 + 2.21T + 3T^{2} \) |
| 5 | \( 1 - 3.59T + 5T^{2} \) |
| 7 | \( 1 - 5.06T + 7T^{2} \) |
| 11 | \( 1 - 2.55T + 11T^{2} \) |
| 13 | \( 1 - 4.93T + 13T^{2} \) |
| 17 | \( 1 + 2.68T + 17T^{2} \) |
| 19 | \( 1 + 4.72T + 19T^{2} \) |
| 23 | \( 1 + 1.49T + 23T^{2} \) |
| 29 | \( 1 + 2.43T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 - 2.90T + 37T^{2} \) |
| 43 | \( 1 + 7.62T + 43T^{2} \) |
| 47 | \( 1 + 5.15T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 - 1.49T + 59T^{2} \) |
| 61 | \( 1 - 1.06T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 - 8.28T + 73T^{2} \) |
| 79 | \( 1 - 4.28T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 9.36T + 89T^{2} \) |
| 97 | \( 1 - 3.37T + 97T^{2} \) |
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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
−8.193336753382257660749269551140, −6.93538673515830858185224155073, −6.32686793550529663790725564068, −5.92921898557324499568839671137, −5.21971392893576506600918901737, −4.68537200409089096738632466319, −3.89458075633052435173962312151, −2.29014701559842378351435942418, −1.66649080093027739177911646094, −0.983130730068416537128204228536,
0.983130730068416537128204228536, 1.66649080093027739177911646094, 2.29014701559842378351435942418, 3.89458075633052435173962312151, 4.68537200409089096738632466319, 5.21971392893576506600918901737, 5.92921898557324499568839671137, 6.32686793550529663790725564068, 6.93538673515830858185224155073, 8.193336753382257660749269551140