| L(s) = 1 | + 5.74e5·3-s − 8.41e7·5-s − 5.95e9·7-s + 2.35e11·9-s − 1.24e12·11-s − 7.21e12·13-s − 4.83e13·15-s − 6.50e13·17-s + 5.56e14·19-s − 3.42e15·21-s − 3.08e15·23-s − 4.84e15·25-s + 8.12e16·27-s + 4.69e16·29-s + 2.40e17·31-s − 7.13e17·33-s + 5.01e17·35-s + 5.28e17·37-s − 4.14e18·39-s + 1.59e18·41-s − 7.35e18·43-s − 1.98e19·45-s − 1.63e19·47-s + 8.12e18·49-s − 3.73e19·51-s + 2.64e19·53-s + 1.04e20·55-s + ⋯ |
| L(s) = 1 | + 1.87·3-s − 0.770·5-s − 1.13·7-s + 2.50·9-s − 1.31·11-s − 1.11·13-s − 1.44·15-s − 0.460·17-s + 1.09·19-s − 2.13·21-s − 0.674·23-s − 0.406·25-s + 2.81·27-s + 0.715·29-s + 1.70·31-s − 2.45·33-s + 0.877·35-s + 0.488·37-s − 2.08·39-s + 0.453·41-s − 1.20·43-s − 1.92·45-s − 0.967·47-s + 0.296·49-s − 0.862·51-s + 0.391·53-s + 1.01·55-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
| \(L(12)\) |
\(\approx\) |
\(2.675856056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.675856056\) |
| \(L(\frac{25}{2})\) |
|
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 | \( 1 - 5.74e5T + 9.41e10T^{2} \) |
| 5 | \( 1 + 8.41e7T + 1.19e16T^{2} \) |
| 7 | \( 1 + 5.95e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 1.24e12T + 8.95e23T^{2} \) |
| 13 | \( 1 + 7.21e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 6.50e13T + 1.99e28T^{2} \) |
| 19 | \( 1 - 5.56e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 3.08e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 4.69e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 2.40e17T + 2.00e34T^{2} \) |
| 37 | \( 1 - 5.28e17T + 1.17e36T^{2} \) |
| 41 | \( 1 - 1.59e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 7.35e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 1.63e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 2.64e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 3.24e20T + 5.36e40T^{2} \) |
| 61 | \( 1 - 3.22e20T + 1.15e41T^{2} \) |
| 67 | \( 1 - 1.09e21T + 9.99e41T^{2} \) |
| 71 | \( 1 + 2.30e20T + 3.79e42T^{2} \) |
| 73 | \( 1 - 8.33e20T + 7.18e42T^{2} \) |
| 79 | \( 1 - 2.59e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 1.05e22T + 1.37e44T^{2} \) |
| 89 | \( 1 - 4.07e22T + 6.85e44T^{2} \) |
| 97 | \( 1 + 9.40e22T + 4.96e45T^{2} \) |
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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
−10.05473151662394759137075003699, −9.687229217374254389805660519906, −8.304776669425614081427103243146, −7.74641307228228650665987946548, −6.77031558602377535271361143644, −4.84263125186541410328879841937, −3.70671447456024022490661344025, −2.88773643609696315445957984546, −2.27672396622963242365230790164, −0.58054141752189685504849373118,
0.58054141752189685504849373118, 2.27672396622963242365230790164, 2.88773643609696315445957984546, 3.70671447456024022490661344025, 4.84263125186541410328879841937, 6.77031558602377535271361143644, 7.74641307228228650665987946548, 8.304776669425614081427103243146, 9.687229217374254389805660519906, 10.05473151662394759137075003699
