| L(s) = 1 | − 1.77e5·3-s + 7.43e7·5-s − 7.86e9·7-s + 3.13e10·9-s + 2.75e10·11-s + 3.75e12·13-s − 1.31e13·15-s − 4.00e13·17-s − 4.20e14·19-s + 1.39e15·21-s + 6.48e15·23-s − 6.39e15·25-s − 5.55e15·27-s + 9.22e16·29-s + 8.64e16·31-s − 4.88e15·33-s − 5.84e17·35-s + 4.85e17·37-s − 6.64e17·39-s − 1.96e18·41-s − 1.03e19·43-s + 2.33e18·45-s − 2.48e19·47-s + 3.44e19·49-s + 7.10e18·51-s + 1.27e20·53-s + 2.05e18·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.680·5-s − 1.50·7-s + 0.333·9-s + 0.0291·11-s + 0.580·13-s − 0.392·15-s − 0.283·17-s − 0.828·19-s + 0.867·21-s + 1.41·23-s − 0.536·25-s − 0.192·27-s + 1.40·29-s + 0.610·31-s − 0.0168·33-s − 1.02·35-s + 0.448·37-s − 0.335·39-s − 0.556·41-s − 1.69·43-s + 0.226·45-s − 1.46·47-s + 1.26·49-s + 0.163·51-s + 1.89·53-s + 0.0198·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.77e5T \) |
| good | 5 | \( 1 - 7.43e7T + 1.19e16T^{2} \) |
| 7 | \( 1 + 7.86e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 2.75e10T + 8.95e23T^{2} \) |
| 13 | \( 1 - 3.75e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 4.00e13T + 1.99e28T^{2} \) |
| 19 | \( 1 + 4.20e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 6.48e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 9.22e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 8.64e16T + 2.00e34T^{2} \) |
| 37 | \( 1 - 4.85e17T + 1.17e36T^{2} \) |
| 41 | \( 1 + 1.96e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 1.03e19T + 3.71e37T^{2} \) |
| 47 | \( 1 + 2.48e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 1.27e20T + 4.55e39T^{2} \) |
| 59 | \( 1 - 1.71e20T + 5.36e40T^{2} \) |
| 61 | \( 1 - 1.70e20T + 1.15e41T^{2} \) |
| 67 | \( 1 - 1.93e20T + 9.99e41T^{2} \) |
| 71 | \( 1 - 2.56e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 3.27e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 8.02e20T + 4.42e43T^{2} \) |
| 83 | \( 1 - 1.51e22T + 1.37e44T^{2} \) |
| 89 | \( 1 - 2.67e22T + 6.85e44T^{2} \) |
| 97 | \( 1 + 6.15e21T + 4.96e45T^{2} \) |
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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
−10.46382949529010457408390611782, −9.725416152376083742178868933080, −8.574407489763354542673277657102, −6.71820519319299600947633483069, −6.31290142683041832080976518021, −5.05388560998067029299756540470, −3.64278581947448121234224742718, −2.49691945237211735945236914476, −1.08912403117671560628325864148, 0,
1.08912403117671560628325864148, 2.49691945237211735945236914476, 3.64278581947448121234224742718, 5.05388560998067029299756540470, 6.31290142683041832080976518021, 6.71820519319299600947633483069, 8.574407489763354542673277657102, 9.725416152376083742178868933080, 10.46382949529010457408390611782
