| L(s) = 1 | − 2-s + 4-s + 2.82i·5-s + 4.24i·7-s − 8-s − 2.82i·10-s + 5.65i·13-s − 4.24i·14-s + 16-s + 4·17-s + 5.65i·19-s + 2.82i·20-s − 1.41i·23-s − 3.00·25-s − 5.65i·26-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.26i·5-s + 1.60i·7-s − 0.353·8-s − 0.894i·10-s + 1.56i·13-s − 1.13i·14-s + 0.250·16-s + 0.970·17-s + 1.29i·19-s + 0.632i·20-s − 0.294i·23-s − 0.600·25-s − 1.10i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.298914858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.298914858\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 7 | \( 1 - 4.24iT - 7T^{2} \) |
| 13 | \( 1 - 5.65iT - 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 5.65iT - 19T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 1.41iT - 47T^{2} \) |
| 53 | \( 1 + 2.82iT - 53T^{2} \) |
| 59 | \( 1 - 14.1iT - 59T^{2} \) |
| 61 | \( 1 + 8.48iT - 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 12.7iT - 71T^{2} \) |
| 73 | \( 1 - 1.41iT - 73T^{2} \) |
| 79 | \( 1 + 7.07iT - 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 18.3iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.464720148285383274395083211133, −8.591719854503404647973526606440, −7.994022171981303662318707903022, −7.03432867991204954792087472939, −6.28591626832989856474494922376, −5.86058063485458486945352480239, −4.56586817801460336750958268308, −3.25212140046988975363211780834, −2.56905041531577077988028504277, −1.64879463858937532460343046813,
0.76326666849079354093271603211, 0.981928877622687933044407594463, 2.76071446101906306769807303111, 3.81068719113730444364575991074, 4.79266877933545202269196340713, 5.45735686938429888370837138293, 6.61932953362489900343342617688, 7.42427370593395791912102636538, 8.060706767888440493666686375696, 8.600506626386974754733153599815
