| L(s) = 1 | − 2.30e3·2-s + 5.86e5·3-s − 3.09e6·4-s + 4.69e7·5-s − 1.34e9·6-s + 2.64e10·8-s + 2.49e11·9-s − 1.07e11·10-s − 1.75e12·11-s − 1.81e12·12-s − 1.84e12·13-s + 2.75e13·15-s − 3.48e13·16-s + 1.02e14·17-s − 5.74e14·18-s + 4.77e14·19-s − 1.45e14·20-s + 4.02e15·22-s − 2.32e15·23-s + 1.54e16·24-s − 9.71e15·25-s + 4.23e15·26-s + 9.12e16·27-s − 4.91e15·29-s − 6.33e16·30-s − 1.44e15·31-s − 1.41e17·32-s + ⋯ |
| L(s) = 1 | − 0.794·2-s + 1.91·3-s − 0.368·4-s + 0.429·5-s − 1.51·6-s + 1.08·8-s + 2.65·9-s − 0.341·10-s − 1.84·11-s − 0.705·12-s − 0.285·13-s + 0.821·15-s − 0.495·16-s + 0.724·17-s − 2.10·18-s + 0.940·19-s − 0.158·20-s + 1.46·22-s − 0.509·23-s + 2.07·24-s − 0.815·25-s + 0.226·26-s + 3.15·27-s − 0.0747·29-s − 0.652·30-s − 0.0102·31-s − 0.694·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(2.925098480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.925098480\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 2.30e3T + 8.38e6T^{2} \) |
| 3 | \( 1 - 5.86e5T + 9.41e10T^{2} \) |
| 5 | \( 1 - 4.69e7T + 1.19e16T^{2} \) |
| 11 | \( 1 + 1.75e12T + 8.95e23T^{2} \) |
| 13 | \( 1 + 1.84e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 1.02e14T + 1.99e28T^{2} \) |
| 19 | \( 1 - 4.77e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 2.32e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 4.91e15T + 4.31e33T^{2} \) |
| 31 | \( 1 + 1.44e15T + 2.00e34T^{2} \) |
| 37 | \( 1 - 4.58e17T + 1.17e36T^{2} \) |
| 41 | \( 1 - 1.00e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 5.74e18T + 3.71e37T^{2} \) |
| 47 | \( 1 - 2.44e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 5.05e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 5.09e19T + 5.36e40T^{2} \) |
| 61 | \( 1 + 5.78e19T + 1.15e41T^{2} \) |
| 67 | \( 1 + 8.23e19T + 9.99e41T^{2} \) |
| 71 | \( 1 - 1.25e21T + 3.79e42T^{2} \) |
| 73 | \( 1 - 2.52e20T + 7.18e42T^{2} \) |
| 79 | \( 1 - 2.90e20T + 4.42e43T^{2} \) |
| 83 | \( 1 - 1.03e22T + 1.37e44T^{2} \) |
| 89 | \( 1 - 1.40e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 3.46e22T + 4.96e45T^{2} \) |
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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
−10.37373810326239004259615782449, −9.805772739863419477437508761397, −8.948170222412823473027160560960, −7.78093239847906611154262042993, −7.61997677072175090779548981243, −5.29387019646410041572394279363, −3.99341281755490891640004099356, −2.78130029953167305521141153907, −1.98806666419809905481974502412, −0.77505849649339752481579077043,
0.77505849649339752481579077043, 1.98806666419809905481974502412, 2.78130029953167305521141153907, 3.99341281755490891640004099356, 5.29387019646410041572394279363, 7.61997677072175090779548981243, 7.78093239847906611154262042993, 8.948170222412823473027160560960, 9.805772739863419477437508761397, 10.37373810326239004259615782449
