| L(s) = 1 | − 3.87e5·3-s + 2.09e8·5-s + 7.31e8·7-s + 5.57e10·9-s + 1.52e12·11-s − 3.20e12·13-s − 8.09e13·15-s − 1.42e14·17-s + 3.73e14·19-s − 2.83e14·21-s − 1.74e14·23-s + 3.18e16·25-s + 1.48e16·27-s + 6.76e16·29-s + 1.20e17·31-s − 5.90e17·33-s + 1.53e17·35-s − 1.03e18·37-s + 1.24e18·39-s + 4.44e18·41-s + 3.59e18·43-s + 1.16e19·45-s + 1.37e19·47-s − 2.68e19·49-s + 5.52e19·51-s − 1.46e19·53-s + 3.18e20·55-s + ⋯ |
| L(s) = 1 | − 1.26·3-s + 1.91·5-s + 0.139·7-s + 0.592·9-s + 1.61·11-s − 0.496·13-s − 2.41·15-s − 1.00·17-s + 0.736·19-s − 0.176·21-s − 0.0381·23-s + 2.67·25-s + 0.514·27-s + 1.02·29-s + 0.849·31-s − 2.03·33-s + 0.268·35-s − 0.960·37-s + 0.626·39-s + 1.26·41-s + 0.589·43-s + 1.13·45-s + 0.812·47-s − 0.980·49-s + 1.27·51-s − 0.217·53-s + 3.08·55-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
| \(L(12)\) |
\(\approx\) |
\(2.682853945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.682853945\) |
| \(L(\frac{25}{2})\) |
|
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 | \( 1 + 3.87e5T + 9.41e10T^{2} \) |
| 5 | \( 1 - 2.09e8T + 1.19e16T^{2} \) |
| 7 | \( 1 - 7.31e8T + 2.73e19T^{2} \) |
| 11 | \( 1 - 1.52e12T + 8.95e23T^{2} \) |
| 13 | \( 1 + 3.20e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 1.42e14T + 1.99e28T^{2} \) |
| 19 | \( 1 - 3.73e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 1.74e14T + 2.08e31T^{2} \) |
| 29 | \( 1 - 6.76e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 1.20e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.03e18T + 1.17e36T^{2} \) |
| 41 | \( 1 - 4.44e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 3.59e18T + 3.71e37T^{2} \) |
| 47 | \( 1 - 1.37e19T + 2.87e38T^{2} \) |
| 53 | \( 1 + 1.46e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 4.80e19T + 5.36e40T^{2} \) |
| 61 | \( 1 - 1.58e20T + 1.15e41T^{2} \) |
| 67 | \( 1 - 4.69e20T + 9.99e41T^{2} \) |
| 71 | \( 1 - 1.33e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 2.99e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 5.35e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 1.08e22T + 1.37e44T^{2} \) |
| 89 | \( 1 + 1.80e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 2.49e22T + 4.96e45T^{2} \) |
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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
−10.64794478795969037492909626000, −9.701868196933496544449276628746, −8.896083841649627775781919428626, −6.79873117879802028788637481171, −6.28313621307789336993760391813, −5.40463486895931924590010446507, −4.48278018458656073835897991348, −2.65167498241424641065728994165, −1.53205376043115271450235796313, −0.78544217276068543147516566702,
0.78544217276068543147516566702, 1.53205376043115271450235796313, 2.65167498241424641065728994165, 4.48278018458656073835897991348, 5.40463486895931924590010446507, 6.28313621307789336993760391813, 6.79873117879802028788637481171, 8.896083841649627775781919428626, 9.701868196933496544449276628746, 10.64794478795969037492909626000
