| L(s) = 1 | − 1.57e3·2-s + 5.90e4·3-s + 3.70e5·4-s − 9.27e7·6-s − 1.39e9·7-s + 2.71e9·8-s + 3.48e9·9-s − 1.07e11·11-s + 2.18e10·12-s − 6.34e11·13-s + 2.19e12·14-s − 5.03e12·16-s + 9.95e12·17-s − 5.47e12·18-s + 2.82e12·19-s − 8.24e13·21-s + 1.69e14·22-s − 7.33e13·23-s + 1.60e14·24-s + 9.95e14·26-s + 2.05e14·27-s − 5.17e14·28-s − 1.24e15·29-s + 6.63e15·31-s + 2.22e15·32-s − 6.36e15·33-s − 1.56e16·34-s + ⋯ |
| L(s) = 1 | − 1.08·2-s + 0.577·3-s + 0.176·4-s − 0.626·6-s − 1.86·7-s + 0.893·8-s + 0.333·9-s − 1.25·11-s + 0.101·12-s − 1.27·13-s + 2.02·14-s − 1.14·16-s + 1.19·17-s − 0.361·18-s + 0.105·19-s − 1.07·21-s + 1.35·22-s − 0.369·23-s + 0.515·24-s + 1.38·26-s + 0.192·27-s − 0.329·28-s − 0.549·29-s + 1.45·31-s + 0.349·32-s − 0.723·33-s − 1.29·34-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 | \( 1 + 1.57e3T + 2.09e6T^{2} \) |
| 7 | \( 1 + 1.39e9T + 5.58e17T^{2} \) |
| 11 | \( 1 + 1.07e11T + 7.40e21T^{2} \) |
| 13 | \( 1 + 6.34e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 9.95e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 2.82e12T + 7.14e26T^{2} \) |
| 23 | \( 1 + 7.33e13T + 3.94e28T^{2} \) |
| 29 | \( 1 + 1.24e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 6.63e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 5.11e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 5.89e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.83e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 4.57e16T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.30e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 5.67e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 7.20e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 2.10e18T + 2.22e38T^{2} \) |
| 71 | \( 1 - 4.72e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 5.09e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 7.07e18T + 7.08e39T^{2} \) |
| 83 | \( 1 - 5.62e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 1.35e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 6.40e19T + 5.27e41T^{2} \) |
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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
−9.742586145352505707840955037681, −9.406095272609600917707076479186, −7.927897401380157887160564466936, −7.40041058183480106363354264799, −6.00738600427572307209896061176, −4.54758538431999645114479129540, −3.12804560981142194178145609063, −2.40898206063045078194774687103, −0.809324354537250670394716227000, 0,
0.809324354537250670394716227000, 2.40898206063045078194774687103, 3.12804560981142194178145609063, 4.54758538431999645114479129540, 6.00738600427572307209896061176, 7.40041058183480106363354264799, 7.927897401380157887160564466936, 9.406095272609600917707076479186, 9.742586145352505707840955037681
