L(s) = 1 | + 2.53e5·3-s + 1.36e8·5-s − 4.69e9·7-s − 3.00e10·9-s + 2.44e11·11-s + 9.00e12·13-s + 3.46e13·15-s + 1.66e14·17-s + 6.92e14·19-s − 1.18e15·21-s + 8.91e13·23-s + 6.81e15·25-s − 3.14e16·27-s − 7.47e16·29-s + 1.11e17·31-s + 6.19e16·33-s − 6.42e17·35-s − 1.38e18·37-s + 2.28e18·39-s − 5.25e17·41-s + 1.20e19·43-s − 4.11e18·45-s − 8.11e18·47-s − 5.34e18·49-s + 4.21e19·51-s + 9.12e19·53-s + 3.34e19·55-s + ⋯ |
L(s) = 1 | + 0.824·3-s + 1.25·5-s − 0.897·7-s − 0.319·9-s + 0.258·11-s + 1.39·13-s + 1.03·15-s + 1.17·17-s + 1.36·19-s − 0.739·21-s + 0.0195·23-s + 0.571·25-s − 1.08·27-s − 1.13·29-s + 0.787·31-s + 0.213·33-s − 1.12·35-s − 1.28·37-s + 1.14·39-s − 0.149·41-s + 1.98·43-s − 0.400·45-s − 0.478·47-s − 0.195·49-s + 0.971·51-s + 1.35·53-s + 0.324·55-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
\(L(12)\) |
\(\approx\) |
\(4.378600845\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.378600845\) |
\(L(\frac{25}{2})\) |
|
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 | \( 1 - 2.53e5T + 9.41e10T^{2} \) |
| 5 | \( 1 - 1.36e8T + 1.19e16T^{2} \) |
| 7 | \( 1 + 4.69e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 2.44e11T + 8.95e23T^{2} \) |
| 13 | \( 1 - 9.00e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 1.66e14T + 1.99e28T^{2} \) |
| 19 | \( 1 - 6.92e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 8.91e13T + 2.08e31T^{2} \) |
| 29 | \( 1 + 7.47e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 1.11e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.38e18T + 1.17e36T^{2} \) |
| 41 | \( 1 + 5.25e17T + 1.24e37T^{2} \) |
| 43 | \( 1 - 1.20e19T + 3.71e37T^{2} \) |
| 47 | \( 1 + 8.11e18T + 2.87e38T^{2} \) |
| 53 | \( 1 - 9.12e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 4.14e20T + 5.36e40T^{2} \) |
| 61 | \( 1 + 6.24e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 3.72e20T + 9.99e41T^{2} \) |
| 71 | \( 1 + 2.77e21T + 3.79e42T^{2} \) |
| 73 | \( 1 - 1.39e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 4.37e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 1.10e21T + 1.37e44T^{2} \) |
| 89 | \( 1 - 5.05e22T + 6.85e44T^{2} \) |
| 97 | \( 1 + 7.16e21T + 4.96e45T^{2} \) |
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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
−10.32780490430107163776848499849, −9.465109178788060710855980919499, −8.799024221528586227064290377064, −7.48906071529783151281292292141, −6.11075874511094559906377326811, −5.55156382274342296091545785224, −3.62886259382546418988883993901, −3.01480932833622032853319404565, −1.84248036700612153862678437219, −0.855969042453938478598309234132,
0.855969042453938478598309234132, 1.84248036700612153862678437219, 3.01480932833622032853319404565, 3.62886259382546418988883993901, 5.55156382274342296091545785224, 6.11075874511094559906377326811, 7.48906071529783151281292292141, 8.799024221528586227064290377064, 9.465109178788060710855980919499, 10.32780490430107163776848499849