| L(s) = 1 | − 6.34·3-s + 1.20i·5-s + 13.2·9-s + 32.0i·11-s − 59.4i·13-s − 7.63i·15-s + 0.0501i·17-s + 11.3·19-s + 75.2i·23-s + 123.·25-s + 87.3·27-s − 263.·29-s + 207.·31-s − 203. i·33-s + 199.·37-s + ⋯ |
| L(s) = 1 | − 1.22·3-s + 0.107i·5-s + 0.489·9-s + 0.878i·11-s − 1.26i·13-s − 0.131i·15-s + 0.000715i·17-s + 0.137·19-s + 0.682i·23-s + 0.988·25-s + 0.622·27-s − 1.68·29-s + 1.20·31-s − 1.07i·33-s + 0.885·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3447400143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3447400143\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 6.34T + 27T^{2} \) |
| 5 | \( 1 - 1.20iT - 125T^{2} \) |
| 11 | \( 1 - 32.0iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 59.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 0.0501iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 11.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 75.2iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 263.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 207.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 199.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 437. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 174. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 283.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 7.41T + 1.48e5T^{2} \) |
| 59 | \( 1 - 38.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 276. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 965. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 580. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 607. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 827. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 1.20e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 312. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.24e3iT - 9.12e5T^{2} \) |
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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
−10.34093959844273311560150911448, −9.669850526905409245527871381884, −8.491755441302239451272844412275, −7.46517993915494244594615151357, −6.72951616292619389023126232169, −5.62666276491897064777857002472, −5.19967549915029372086554680267, −3.99443811057655489315036241552, −2.65034673367429240120476062421, −1.08835850436984641967643357676,
0.13295804866478027931091289844, 1.34751345270782241688080864701, 2.94158109663205612826095797453, 4.33524523658659197387096332701, 5.08707782366450962714878653406, 6.19597055596924962387547537749, 6.54534986414690356703988735252, 7.79747544752350065623813595491, 8.793030987843580939886276416532, 9.606701700509111135792277265909
