| L(s) = 1 | − 3.40e5·3-s + 8.81e7·5-s + 8.68e9·7-s + 2.18e10·9-s − 1.20e12·11-s − 1.25e13·13-s − 3.00e13·15-s + 6.10e13·17-s − 2.20e14·19-s − 2.95e15·21-s − 5.85e15·23-s − 4.14e15·25-s + 2.46e16·27-s + 9.56e16·29-s + 7.90e16·31-s + 4.11e17·33-s + 7.66e17·35-s − 2.09e18·37-s + 4.27e18·39-s + 2.65e18·41-s + 1.82e18·43-s + 1.92e18·45-s − 1.31e19·47-s + 4.81e19·49-s − 2.07e19·51-s − 1.04e19·53-s − 1.06e20·55-s + ⋯ |
| L(s) = 1 | − 1.11·3-s + 0.807·5-s + 1.66·7-s + 0.232·9-s − 1.27·11-s − 1.94·13-s − 0.896·15-s + 0.431·17-s − 0.434·19-s − 1.84·21-s − 1.28·23-s − 0.347·25-s + 0.852·27-s + 1.45·29-s + 0.558·31-s + 1.41·33-s + 1.34·35-s − 1.93·37-s + 2.15·39-s + 0.754·41-s + 0.299·43-s + 0.187·45-s − 0.778·47-s + 1.75·49-s − 0.479·51-s − 0.155·53-s − 1.02·55-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
| \(L(12)\) |
\(\approx\) |
\(1.154720289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.154720289\) |
| \(L(\frac{25}{2})\) |
|
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 | \( 1 + 3.40e5T + 9.41e10T^{2} \) |
| 5 | \( 1 - 8.81e7T + 1.19e16T^{2} \) |
| 7 | \( 1 - 8.68e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 1.20e12T + 8.95e23T^{2} \) |
| 13 | \( 1 + 1.25e13T + 4.17e25T^{2} \) |
| 17 | \( 1 - 6.10e13T + 1.99e28T^{2} \) |
| 19 | \( 1 + 2.20e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 5.85e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 9.56e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 7.90e16T + 2.00e34T^{2} \) |
| 37 | \( 1 + 2.09e18T + 1.17e36T^{2} \) |
| 41 | \( 1 - 2.65e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 1.82e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 1.31e19T + 2.87e38T^{2} \) |
| 53 | \( 1 + 1.04e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 1.26e20T + 5.36e40T^{2} \) |
| 61 | \( 1 - 2.84e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 9.17e20T + 9.99e41T^{2} \) |
| 71 | \( 1 + 1.83e21T + 3.79e42T^{2} \) |
| 73 | \( 1 - 1.84e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 4.20e21T + 4.42e43T^{2} \) |
| 83 | \( 1 + 5.94e21T + 1.37e44T^{2} \) |
| 89 | \( 1 - 3.55e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 2.81e22T + 4.96e45T^{2} \) |
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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
−10.56441024674030963989668405429, −10.04114560431900658232348209278, −8.333303181307878306559158040067, −7.44444517044172570230910730521, −6.02353916680775944596435075319, −5.10000074950542886403023708631, −4.73494908716686709946372597045, −2.54441431868666457967615333876, −1.79085139002705524381810750533, −0.45679590138563647361981447457,
0.45679590138563647361981447457, 1.79085139002705524381810750533, 2.54441431868666457967615333876, 4.73494908716686709946372597045, 5.10000074950542886403023708631, 6.02353916680775944596435075319, 7.44444517044172570230910730521, 8.333303181307878306559158040067, 10.04114560431900658232348209278, 10.56441024674030963989668405429
