| L(s) = 1 | − 2.93i·3-s − 5.62·7-s + 0.402·9-s − 17.4·11-s − 7.35i·13-s + 25.3·17-s + (−2.59 − 18.8i)19-s + 16.4i·21-s − 19.3·23-s − 27.5i·27-s − 4.89i·29-s − 45.2i·31-s + 51.1i·33-s + 25.2i·37-s − 21.5·39-s + ⋯ |
| L(s) = 1 | − 0.977i·3-s − 0.803·7-s + 0.0446·9-s − 1.58·11-s − 0.565i·13-s + 1.48·17-s + (−0.136 − 0.990i)19-s + 0.785i·21-s − 0.839·23-s − 1.02i·27-s − 0.168i·29-s − 1.45i·31-s + 1.55i·33-s + 0.683i·37-s − 0.553·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.05994724170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05994724170\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (2.59 + 18.8i)T \) |
| good | 3 | \( 1 + 2.93iT - 9T^{2} \) |
| 7 | \( 1 + 5.62T + 49T^{2} \) |
| 11 | \( 1 + 17.4T + 121T^{2} \) |
| 13 | \( 1 + 7.35iT - 169T^{2} \) |
| 17 | \( 1 - 25.3T + 289T^{2} \) |
| 23 | \( 1 + 19.3T + 529T^{2} \) |
| 29 | \( 1 + 4.89iT - 841T^{2} \) |
| 31 | \( 1 + 45.2iT - 961T^{2} \) |
| 37 | \( 1 - 25.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 16.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 28.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 38.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 81.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 61.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 94.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 109. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 57.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 36.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 112. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 42.1T + 6.88e3T^{2} \) |
| 89 | \( 1 - 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 148. iT - 9.40e3T^{2} \) |
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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
−9.311771966222494141093546906015, −7.992493218589410618084567021093, −7.83136883058151537381795359251, −6.98886770740572469091637803471, −6.05842375945151860972581756699, −5.47887132006762186416593619771, −4.32697679166734387774016605754, −3.05068642206423488429912489421, −2.43286533692508062093608506036, −1.03942936699320557956763351113,
0.01687750745331200847519195262, 1.78659581747728642756949147617, 3.16927959520252692923139214082, 3.66389896446927688791137662893, 4.79872504743547186962837728048, 5.44140034880253423172046782140, 6.30098110965641311547703837663, 7.38026559935829339924629007886, 8.021028037107968278156571136246, 8.986990148334890469350026677471
