| L(s) = 1 | − 1.77e5·3-s − 2.66e7·5-s − 2.63e9·7-s + 3.13e10·9-s + 6.90e11·11-s + 8.43e12·13-s + 4.72e12·15-s + 1.75e13·17-s + 5.46e14·19-s + 4.67e14·21-s − 4.16e15·23-s − 1.12e16·25-s − 5.55e15·27-s + 7.34e16·29-s − 1.44e17·31-s − 1.22e17·33-s + 7.04e16·35-s + 4.39e16·37-s − 1.49e18·39-s + 1.13e18·41-s + 3.65e18·43-s − 8.37e17·45-s − 1.31e18·47-s − 2.04e19·49-s − 3.10e18·51-s + 1.56e18·53-s − 1.84e19·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.244·5-s − 0.504·7-s + 0.333·9-s + 0.729·11-s + 1.30·13-s + 0.141·15-s + 0.124·17-s + 1.07·19-s + 0.291·21-s − 0.912·23-s − 0.940·25-s − 0.192·27-s + 1.11·29-s − 1.01·31-s − 0.421·33-s + 0.123·35-s + 0.0406·37-s − 0.754·39-s + 0.322·41-s + 0.599·43-s − 0.0815·45-s − 0.0775·47-s − 0.745·49-s − 0.0717·51-s + 0.0232·53-s − 0.178·55-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
| \(L(12)\) |
\(\approx\) |
\(1.665875348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.665875348\) |
| \(L(\frac{25}{2})\) |
|
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 | \( 1 + 2.66e7T + 1.19e16T^{2} \) |
| 7 | \( 1 + 2.63e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 6.90e11T + 8.95e23T^{2} \) |
| 13 | \( 1 - 8.43e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 1.75e13T + 1.99e28T^{2} \) |
| 19 | \( 1 - 5.46e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 4.16e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 7.34e16T + 4.31e33T^{2} \) |
| 31 | \( 1 + 1.44e17T + 2.00e34T^{2} \) |
| 37 | \( 1 - 4.39e16T + 1.17e36T^{2} \) |
| 41 | \( 1 - 1.13e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 3.65e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 1.31e18T + 2.87e38T^{2} \) |
| 53 | \( 1 - 1.56e18T + 4.55e39T^{2} \) |
| 59 | \( 1 + 1.30e20T + 5.36e40T^{2} \) |
| 61 | \( 1 - 2.40e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 2.77e20T + 9.99e41T^{2} \) |
| 71 | \( 1 + 2.66e21T + 3.79e42T^{2} \) |
| 73 | \( 1 - 2.44e20T + 7.18e42T^{2} \) |
| 79 | \( 1 + 4.73e21T + 4.42e43T^{2} \) |
| 83 | \( 1 + 3.82e21T + 1.37e44T^{2} \) |
| 89 | \( 1 - 3.93e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 1.29e23T + 4.96e45T^{2} \) |
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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
−11.32969407640527081114628056397, −10.13670193990219090791829325673, −9.052415480327063442192571162613, −7.73582881779698906397335370164, −6.48468885171220654976367938535, −5.68411966853243079414132143102, −4.19799140560388403043158303669, −3.30148728135651030914816267767, −1.61711205176281680364924757074, −0.61080094403075609416582109145,
0.61080094403075609416582109145, 1.61711205176281680364924757074, 3.30148728135651030914816267767, 4.19799140560388403043158303669, 5.68411966853243079414132143102, 6.48468885171220654976367938535, 7.73582881779698906397335370164, 9.052415480327063442192571162613, 10.13670193990219090791829325673, 11.32969407640527081114628056397
