| L(s) = 1 | − 8i·2-s − 39.9·3-s − 64·4-s − 323. i·5-s + 319. i·6-s − 568. i·7-s + 512i·8-s − 588.·9-s − 2.58e3·10-s + 238. i·11-s + 2.55e3·12-s − 4.54e3·14-s + 1.29e4i·15-s + 4.09e3·16-s − 2.04e4·17-s + 4.70e3i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.855·3-s − 0.5·4-s − 1.15i·5-s + 0.604i·6-s − 0.626i·7-s + 0.353i·8-s − 0.268·9-s − 0.818·10-s + 0.0539i·11-s + 0.427·12-s − 0.443·14-s + 0.990i·15-s + 0.250·16-s − 1.01·17-s + 0.190i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.8619610409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8619610409\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8iT \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 39.9T + 2.18e3T^{2} \) |
| 5 | \( 1 + 323. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 568. iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 238. iT - 1.94e7T^{2} \) |
| 17 | \( 1 + 2.04e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 9.64e3iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 7.82e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.38e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.60e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 1.52e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 1.85e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 + 8.50e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.20e6iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 6.65e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.48e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 3.04e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.87e5iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 3.68e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + 1.57e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 2.29e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.93e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 8.15e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 + 1.33e6iT - 8.07e13T^{2} \) |
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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
−10.50909254446312383968265436834, −9.199215587392234387577152143507, −8.700732420312511108914972053852, −7.34892037427294304797794247441, −6.11833692512371083802136061119, −4.97212976653520019603767483783, −4.48330134204245843503117292351, −3.00923689381029549655105776521, −1.43293200281877317531790896554, −0.61387212433607916546038049633,
0.36353430297368156699168250972, 2.25677612196461747525294693063, 3.47110058126460420576348927063, 4.92440304809026562595712609931, 5.82787707316120928387479520525, 6.55327912886416284384544177417, 7.33837318659766354421136572878, 8.576096320038673397667656923542, 9.471158897555456492744624305880, 10.76771826858610253369946921657
