| L(s) = 1 | − 1.46e3·2-s − 5.90e4·3-s + 4.62e4·4-s + 8.64e7·6-s − 1.40e8·7-s + 3.00e9·8-s + 3.48e9·9-s + 4.97e9·11-s − 2.73e9·12-s + 4.81e11·13-s + 2.05e11·14-s − 4.49e12·16-s − 5.10e12·17-s − 5.10e12·18-s + 3.13e13·19-s + 8.29e12·21-s − 7.28e12·22-s − 2.01e14·23-s − 1.77e14·24-s − 7.04e14·26-s − 2.05e14·27-s − 6.49e12·28-s + 2.45e14·29-s − 1.57e15·31-s + 2.80e14·32-s − 2.93e14·33-s + 7.46e15·34-s + ⋯ |
| L(s) = 1 | − 1.01·2-s − 0.577·3-s + 0.0220·4-s + 0.583·6-s − 0.187·7-s + 0.988·8-s + 0.333·9-s + 0.0578·11-s − 0.0127·12-s + 0.967·13-s + 0.189·14-s − 1.02·16-s − 0.613·17-s − 0.336·18-s + 1.17·19-s + 0.108·21-s − 0.0584·22-s − 1.01·23-s − 0.570·24-s − 0.978·26-s − 0.192·27-s − 0.00414·28-s + 0.108·29-s − 0.344·31-s + 0.0441·32-s − 0.0333·33-s + 0.620·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 5.90e4T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.46e3T + 2.09e6T^{2} \) |
| 7 | \( 1 + 1.40e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 4.97e9T + 7.40e21T^{2} \) |
| 13 | \( 1 - 4.81e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 5.10e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 3.13e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 2.01e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 2.45e14T + 5.13e30T^{2} \) |
| 31 | \( 1 + 1.57e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 3.62e15T + 8.55e32T^{2} \) |
| 41 | \( 1 + 1.14e17T + 7.38e33T^{2} \) |
| 43 | \( 1 + 3.06e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 4.22e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 6.40e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 3.11e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 1.70e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 7.05e18T + 2.22e38T^{2} \) |
| 71 | \( 1 - 1.57e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 2.49e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.06e20T + 7.08e39T^{2} \) |
| 83 | \( 1 - 8.98e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 1.82e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 7.66e20T + 5.27e41T^{2} \) |
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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
−10.03573705582816865757875068187, −9.110957428423687395838113557592, −8.122616309423196460116335206602, −7.05623024851108630440136858601, −5.91250728181516435302999842650, −4.70288980015461378718692912918, −3.55777017047490422171724713475, −1.86347746446649611117619491840, −0.927648764139489564594197663028, 0,
0.927648764139489564594197663028, 1.86347746446649611117619491840, 3.55777017047490422171724713475, 4.70288980015461378718692912918, 5.91250728181516435302999842650, 7.05623024851108630440136858601, 8.122616309423196460116335206602, 9.110957428423687395838113557592, 10.03573705582816865757875068187
