| L(s) = 1 | + 1.86e3·2-s − 3.00e7·4-s − 6.53e8·5-s − 4.71e10·7-s − 1.18e11·8-s − 1.21e12·10-s − 1.60e13·11-s + 1.23e14·13-s − 8.79e13·14-s + 7.88e14·16-s − 1.37e15·17-s + 2.03e14·19-s + 1.96e16·20-s − 2.99e16·22-s − 1.96e14·23-s + 1.28e17·25-s + 2.29e17·26-s + 1.41e18·28-s − 1.83e18·29-s − 4.66e18·31-s + 5.44e18·32-s − 2.57e18·34-s + 3.08e19·35-s + 4.09e19·37-s + 3.79e17·38-s + 7.74e19·40-s + 5.64e19·41-s + ⋯ |
| L(s) = 1 | + 0.321·2-s − 0.896·4-s − 1.19·5-s − 1.28·7-s − 0.610·8-s − 0.385·10-s − 1.54·11-s + 1.46·13-s − 0.414·14-s + 0.700·16-s − 0.574·17-s + 0.0211·19-s + 1.07·20-s − 0.496·22-s − 0.00186·23-s + 0.431·25-s + 0.472·26-s + 1.15·28-s − 0.963·29-s − 1.06·31-s + 0.835·32-s − 0.184·34-s + 1.54·35-s + 1.02·37-s + 0.00679·38-s + 0.730·40-s + 0.390·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(0.4699619289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4699619289\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 1.86e3T + 3.35e7T^{2} \) |
| 5 | \( 1 + 6.53e8T + 2.98e17T^{2} \) |
| 7 | \( 1 + 4.71e10T + 1.34e21T^{2} \) |
| 11 | \( 1 + 1.60e13T + 1.08e26T^{2} \) |
| 13 | \( 1 - 1.23e14T + 7.05e27T^{2} \) |
| 17 | \( 1 + 1.37e15T + 5.77e30T^{2} \) |
| 19 | \( 1 - 2.03e14T + 9.30e31T^{2} \) |
| 23 | \( 1 + 1.96e14T + 1.10e34T^{2} \) |
| 29 | \( 1 + 1.83e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 4.66e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 4.09e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 5.64e19T + 2.08e40T^{2} \) |
| 43 | \( 1 - 1.41e20T + 6.86e40T^{2} \) |
| 47 | \( 1 - 1.37e21T + 6.34e41T^{2} \) |
| 53 | \( 1 + 4.94e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 3.33e21T + 1.86e44T^{2} \) |
| 61 | \( 1 + 2.98e22T + 4.29e44T^{2} \) |
| 67 | \( 1 - 3.37e22T + 4.48e45T^{2} \) |
| 71 | \( 1 + 1.69e23T + 1.91e46T^{2} \) |
| 73 | \( 1 + 2.16e23T + 3.82e46T^{2} \) |
| 79 | \( 1 - 4.45e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 1.39e24T + 9.48e47T^{2} \) |
| 89 | \( 1 + 1.32e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + 5.44e24T + 4.66e49T^{2} \) |
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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
−15.36476701081213256901345340694, −13.43111455840707688865728528112, −12.66587085628942664878716721025, −10.85495472046424028287799765993, −9.139544031113178198750872578739, −7.76985995529455777981801587896, −5.85296802424939885154673520510, −4.14613679868101772296265438649, −3.14817286772350203942095806573, −0.37994093876519177585829330473,
0.37994093876519177585829330473, 3.14817286772350203942095806573, 4.14613679868101772296265438649, 5.85296802424939885154673520510, 7.76985995529455777981801587896, 9.139544031113178198750872578739, 10.85495472046424028287799765993, 12.66587085628942664878716721025, 13.43111455840707688865728528112, 15.36476701081213256901345340694
