| L(s) = 1 | − 4.89e4·3-s + 3.19e7·5-s + 5.17e9·7-s − 9.17e10·9-s + 6.04e11·11-s − 7.96e12·13-s − 1.56e12·15-s + 1.98e13·17-s + 6.27e14·19-s − 2.53e14·21-s + 4.55e15·23-s − 1.08e16·25-s + 9.10e15·27-s − 4.14e16·29-s − 1.35e15·31-s − 2.96e16·33-s + 1.65e17·35-s − 3.41e17·37-s + 3.90e17·39-s − 3.69e18·41-s − 1.96e18·43-s − 2.93e18·45-s − 2.44e19·47-s − 6.25e17·49-s − 9.71e17·51-s + 6.39e19·53-s + 1.93e19·55-s + ⋯ |
| L(s) = 1 | − 0.159·3-s + 0.293·5-s + 0.988·7-s − 0.974·9-s + 0.638·11-s − 1.23·13-s − 0.0467·15-s + 0.140·17-s + 1.23·19-s − 0.157·21-s + 0.997·23-s − 0.914·25-s + 0.315·27-s − 0.630·29-s − 0.00959·31-s − 0.101·33-s + 0.289·35-s − 0.315·37-s + 0.196·39-s − 1.04·41-s − 0.323·43-s − 0.285·45-s − 1.44·47-s − 0.0228·49-s − 0.0224·51-s + 0.947·53-s + 0.187·55-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
| \(L(12)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 4.89e4T + 9.41e10T^{2} \) |
| 5 | \( 1 - 3.19e7T + 1.19e16T^{2} \) |
| 7 | \( 1 - 5.17e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 6.04e11T + 8.95e23T^{2} \) |
| 13 | \( 1 + 7.96e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 1.98e13T + 1.99e28T^{2} \) |
| 19 | \( 1 - 6.27e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 4.55e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 4.14e16T + 4.31e33T^{2} \) |
| 31 | \( 1 + 1.35e15T + 2.00e34T^{2} \) |
| 37 | \( 1 + 3.41e17T + 1.17e36T^{2} \) |
| 41 | \( 1 + 3.69e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 1.96e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 2.44e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 6.39e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 2.81e20T + 5.36e40T^{2} \) |
| 61 | \( 1 - 4.67e20T + 1.15e41T^{2} \) |
| 67 | \( 1 - 2.77e20T + 9.99e41T^{2} \) |
| 71 | \( 1 + 2.29e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 4.56e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 3.99e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 1.45e22T + 1.37e44T^{2} \) |
| 89 | \( 1 - 1.80e21T + 6.85e44T^{2} \) |
| 97 | \( 1 - 8.25e22T + 4.96e45T^{2} \) |
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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
−10.08531613270476264631197543356, −9.072644867017902083380598720847, −7.972344528036566289398435306604, −6.93015639305829947630650925503, −5.53019369799135014166152448687, −4.91247447846222624710035446709, −3.43328400577464420303536908336, −2.26029807067196141916295942649, −1.22718740303448194825563299777, 0,
1.22718740303448194825563299777, 2.26029807067196141916295942649, 3.43328400577464420303536908336, 4.91247447846222624710035446709, 5.53019369799135014166152448687, 6.93015639305829947630650925503, 7.972344528036566289398435306604, 9.072644867017902083380598720847, 10.08531613270476264631197543356
