| L(s) = 1 | − 4i·2-s + 3.71i·3-s − 16·4-s + 14.8·6-s − 10.2i·7-s + 64i·8-s + 229.·9-s + 197.·11-s − 59.4i·12-s − 169i·13-s − 41.0·14-s + 256·16-s − 949. i·17-s − 916. i·18-s − 2.23e3·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.238i·3-s − 0.5·4-s + 0.168·6-s − 0.0791i·7-s + 0.353i·8-s + 0.943·9-s + 0.491·11-s − 0.119i·12-s − 0.277i·13-s − 0.0559·14-s + 0.250·16-s − 0.797i·17-s − 0.666i·18-s − 1.41·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.803941768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.803941768\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + 169iT \) |
| good | 3 | \( 1 - 3.71iT - 243T^{2} \) |
| 7 | \( 1 + 10.2iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 197.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 949. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.23e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 367. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 6.60e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.91e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.39e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 2.05e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.63e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.35e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 3.50e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 6.17e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.37e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.21e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 3.83e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.99e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 8.96e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.81e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 7.82e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.05e4iT - 8.58e9T^{2} \) |
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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
−9.484984370860072076282322543736, −8.994315736877392583497952689137, −7.79426425534590172251355276095, −6.90333102816494350149405095376, −5.76425401034089152965326611079, −4.54960975823399968666830598478, −3.96658780277569106644300777851, −2.71539693346077122485768156496, −1.58768049422526518156796549212, −0.44409259208751514494333250563,
1.02559934351973872989535480549, 2.17883960659460888311907623577, 3.86125847059985752723520927594, 4.49463412409470356696621570470, 5.77717998378022361716435020857, 6.61958254984071574936630875872, 7.25022698745281790057090244978, 8.327843018468920554763025387653, 8.963840970674933210702280936329, 10.01490965016339175430817571516
