L(s) = 1 | − 2.57·2-s − 88.6i·3-s − 121.·4-s + 321. i·5-s + 228. i·6-s + 569. i·7-s + 641.·8-s − 5.66e3·9-s − 828. i·10-s − 4.67e3i·11-s + 1.07e4i·12-s − 7.57e3·13-s − 1.46e3i·14-s + 2.85e4·15-s + 1.38e4·16-s + (−1.78e4 − 9.55e3i)17-s + ⋯ |
L(s) = 1 | − 0.227·2-s − 1.89i·3-s − 0.948·4-s + 1.15i·5-s + 0.431i·6-s + 0.627i·7-s + 0.443·8-s − 2.59·9-s − 0.262i·10-s − 1.05i·11-s + 1.79i·12-s − 0.956·13-s − 0.142i·14-s + 2.18·15-s + 0.847·16-s + (−0.881 − 0.471i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.0649153 + 0.258930i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0649153 + 0.258930i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (1.78e4 + 9.55e3i)T \) |
good | 2 | \( 1 + 2.57T + 128T^{2} \) |
| 3 | \( 1 + 88.6iT - 2.18e3T^{2} \) |
| 5 | \( 1 - 321. iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 569. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 4.67e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 7.57e3T + 6.27e7T^{2} \) |
| 19 | \( 1 + 3.17e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 9.40e3iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 2.86e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + 1.36e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 + 4.22e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 1.40e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 4.50e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.96e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.69e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 5.50e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.70e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 3.41e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.13e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + 3.30e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 2.00e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + 8.55e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.45e4T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.11e6iT - 8.07e13T^{2} \) |
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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
−17.22409764846973193707838701874, −14.64868266429978569125399720686, −13.72891421528669621318394061815, −12.59923920737767883332617744522, −11.13726614010925519113375748250, −8.834047128078352738338718948165, −7.47982335834818396942097798873, −6.04070305908466817144100765063, −2.51525513794626326949820602634, −0.17226911251707662695759496085,
4.29331355207883854402684811028, 4.87590344811477354480675188213, 8.546017331942757774789766732040, 9.534975299104574447175570195146, 10.51861312201993638455024333259, 12.64598198585030929812353852078, 14.33307038881247547514190003587, 15.51872081065884163937180157201, 17.01226934078969433638134885090, 17.24733376220566310883296183192