| L(s) = 1 | + 9.52e4·3-s − 5.05e7·5-s − 3.45e9·7-s − 8.50e10·9-s − 1.58e12·11-s + 7.62e12·13-s − 4.81e12·15-s + 2.45e14·17-s − 2.60e14·19-s − 3.28e14·21-s − 7.85e15·23-s − 9.36e15·25-s − 1.70e16·27-s − 4.59e16·29-s − 4.84e16·31-s − 1.50e17·33-s + 1.74e17·35-s − 6.75e17·37-s + 7.26e17·39-s + 3.34e18·41-s − 6.18e18·43-s + 4.29e18·45-s − 1.60e19·47-s − 1.54e19·49-s + 2.34e19·51-s − 1.07e20·53-s + 7.99e19·55-s + ⋯ |
| L(s) = 1 | + 0.310·3-s − 0.462·5-s − 0.660·7-s − 0.903·9-s − 1.67·11-s + 1.18·13-s − 0.143·15-s + 1.73·17-s − 0.512·19-s − 0.204·21-s − 1.71·23-s − 0.785·25-s − 0.590·27-s − 0.699·29-s − 0.342·31-s − 0.519·33-s + 0.305·35-s − 0.624·37-s + 0.366·39-s + 0.949·41-s − 1.01·43-s + 0.418·45-s − 0.949·47-s − 0.564·49-s + 0.539·51-s − 1.60·53-s + 0.774·55-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
| \(L(12)\) |
\(\approx\) |
\(0.6078220367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6078220367\) |
| \(L(\frac{25}{2})\) |
|
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 | \( 1 - 9.52e4T + 9.41e10T^{2} \) |
| 5 | \( 1 + 5.05e7T + 1.19e16T^{2} \) |
| 7 | \( 1 + 3.45e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 1.58e12T + 8.95e23T^{2} \) |
| 13 | \( 1 - 7.62e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 2.45e14T + 1.99e28T^{2} \) |
| 19 | \( 1 + 2.60e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 7.85e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 4.59e16T + 4.31e33T^{2} \) |
| 31 | \( 1 + 4.84e16T + 2.00e34T^{2} \) |
| 37 | \( 1 + 6.75e17T + 1.17e36T^{2} \) |
| 41 | \( 1 - 3.34e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 6.18e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 1.60e19T + 2.87e38T^{2} \) |
| 53 | \( 1 + 1.07e20T + 4.55e39T^{2} \) |
| 59 | \( 1 - 1.50e20T + 5.36e40T^{2} \) |
| 61 | \( 1 - 3.18e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 4.03e20T + 9.99e41T^{2} \) |
| 71 | \( 1 + 2.40e21T + 3.79e42T^{2} \) |
| 73 | \( 1 - 1.01e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 5.66e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 2.07e22T + 1.37e44T^{2} \) |
| 89 | \( 1 + 2.24e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 4.35e22T + 4.96e45T^{2} \) |
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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
−10.60608478028403808657781318476, −9.648471285721107400022685984163, −8.198565949343337621098788053680, −7.83997932412963727029185811917, −6.14264060183056160530014252249, −5.39362431605444632525785363689, −3.73686089510145425628982393915, −3.10056927521779222962701981382, −1.88673620833691237023699534797, −0.29932874143639524422949041940,
0.29932874143639524422949041940, 1.88673620833691237023699534797, 3.10056927521779222962701981382, 3.73686089510145425628982393915, 5.39362431605444632525785363689, 6.14264060183056160530014252249, 7.83997932412963727029185811917, 8.198565949343337621098788053680, 9.648471285721107400022685984163, 10.60608478028403808657781318476
