| L(s) = 1 | − 4.16·2-s + 17.5·3-s − 14.6·4-s − 72.9·6-s + 220.·7-s + 194.·8-s + 64.1·9-s − 85.7·11-s − 257.·12-s − 1.03e3·13-s − 919.·14-s − 339.·16-s + 313.·17-s − 267.·18-s + 458.·19-s + 3.86e3·21-s + 356.·22-s + 3.44e3·23-s + 3.40e3·24-s + 4.30e3·26-s − 3.13e3·27-s − 3.24e3·28-s − 841·29-s − 7.98e3·31-s − 4.80e3·32-s − 1.50e3·33-s − 1.30e3·34-s + ⋯ |
| L(s) = 1 | − 0.735·2-s + 1.12·3-s − 0.458·4-s − 0.827·6-s + 1.70·7-s + 1.07·8-s + 0.264·9-s − 0.213·11-s − 0.515·12-s − 1.69·13-s − 1.25·14-s − 0.331·16-s + 0.262·17-s − 0.194·18-s + 0.291·19-s + 1.91·21-s + 0.157·22-s + 1.35·23-s + 1.20·24-s + 1.24·26-s − 0.827·27-s − 0.781·28-s − 0.185·29-s − 1.49·31-s − 0.829·32-s − 0.240·33-s − 0.193·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 29 | \( 1 + 841T \) |
| good | 2 | \( 1 + 4.16T + 32T^{2} \) |
| 3 | \( 1 - 17.5T + 243T^{2} \) |
| 7 | \( 1 - 220.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 85.7T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.03e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 313.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 458.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.44e3T + 6.43e6T^{2} \) |
| 31 | \( 1 + 7.98e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 152.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.84e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.07e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.58e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.24e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.43e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.95e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 9.19e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.93e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.69e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.15e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.99e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.30e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.36e4T + 8.58e9T^{2} \) |
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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
−9.097125885623658081249015844029, −8.388519618049684776784568195216, −7.71671699773922738113293566971, −7.25829016203512408371540389941, −5.13651861236746059898230352340, −4.85803432116581246902454607091, −3.48589683825683559379295166634, −2.23528476474178326454498013217, −1.43373176872544598585217631932, 0,
1.43373176872544598585217631932, 2.23528476474178326454498013217, 3.48589683825683559379295166634, 4.85803432116581246902454607091, 5.13651861236746059898230352340, 7.25829016203512408371540389941, 7.71671699773922738113293566971, 8.388519618049684776784568195216, 9.097125885623658081249015844029
