| L(s) = 1 | + 1.95e5·3-s + 1.26e8·5-s + 8.35e9·7-s − 5.59e10·9-s − 2.05e11·11-s + 2.81e12·13-s + 2.47e13·15-s − 2.25e14·17-s − 1.36e14·19-s + 1.63e15·21-s + 4.06e15·23-s + 4.16e15·25-s − 2.93e16·27-s − 1.04e17·29-s + 2.09e17·31-s − 4.01e16·33-s + 1.06e18·35-s + 9.00e17·37-s + 5.50e17·39-s + 5.21e18·41-s + 5.52e18·43-s − 7.09e18·45-s + 6.66e18·47-s + 4.25e19·49-s − 4.41e19·51-s + 9.17e19·53-s − 2.60e19·55-s + ⋯ |
| L(s) = 1 | + 0.636·3-s + 1.16·5-s + 1.59·7-s − 0.594·9-s − 0.217·11-s + 0.436·13-s + 0.739·15-s − 1.59·17-s − 0.268·19-s + 1.01·21-s + 0.889·23-s + 0.349·25-s − 1.01·27-s − 1.59·29-s + 1.47·31-s − 0.138·33-s + 1.85·35-s + 0.831·37-s + 0.277·39-s + 1.48·41-s + 0.906·43-s − 0.690·45-s + 0.392·47-s + 1.55·49-s − 1.01·51-s + 1.35·53-s − 0.252·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(4.626942270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.626942270\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 1.95e5T + 9.41e10T^{2} \) |
| 5 | \( 1 - 1.26e8T + 1.19e16T^{2} \) |
| 7 | \( 1 - 8.35e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 2.05e11T + 8.95e23T^{2} \) |
| 13 | \( 1 - 2.81e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 2.25e14T + 1.99e28T^{2} \) |
| 19 | \( 1 + 1.36e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 4.06e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 1.04e17T + 4.31e33T^{2} \) |
| 31 | \( 1 - 2.09e17T + 2.00e34T^{2} \) |
| 37 | \( 1 - 9.00e17T + 1.17e36T^{2} \) |
| 41 | \( 1 - 5.21e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 5.52e18T + 3.71e37T^{2} \) |
| 47 | \( 1 - 6.66e18T + 2.87e38T^{2} \) |
| 53 | \( 1 - 9.17e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 1.00e20T + 5.36e40T^{2} \) |
| 61 | \( 1 - 3.93e19T + 1.15e41T^{2} \) |
| 67 | \( 1 - 1.31e21T + 9.99e41T^{2} \) |
| 71 | \( 1 - 2.74e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 3.27e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 6.31e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 8.53e21T + 1.37e44T^{2} \) |
| 89 | \( 1 - 1.40e22T + 6.85e44T^{2} \) |
| 97 | \( 1 + 3.66e22T + 4.96e45T^{2} \) |
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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
−10.81058361800326299575628509527, −9.317514689572610842940004160203, −8.637716867713205070864780814619, −7.65143246114053384612464698671, −6.18419771835384839744807942928, −5.22695323855967809275964865738, −4.14811194971656477994155899221, −2.47594905846648770898886752096, −2.06153887308077150986950208979, −0.872749195871355287730559700053,
0.872749195871355287730559700053, 2.06153887308077150986950208979, 2.47594905846648770898886752096, 4.14811194971656477994155899221, 5.22695323855967809275964865738, 6.18419771835384839744807942928, 7.65143246114053384612464698671, 8.637716867713205070864780814619, 9.317514689572610842940004160203, 10.81058361800326299575628509527
