| L(s) = 1 | − i·2-s + 4-s + (−1 + 2i)5-s + i·7-s − 3i·8-s + (2 + i)10-s − 3·11-s + 7i·13-s + 14-s − 16-s + 4i·17-s − 3·19-s + (−1 + 2i)20-s + 3i·22-s + (−3 − 4i)25-s + 7·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.5·4-s + (−0.447 + 0.894i)5-s + 0.377i·7-s − 1.06i·8-s + (0.632 + 0.316i)10-s − 0.904·11-s + 1.94i·13-s + 0.267·14-s − 0.250·16-s + 0.970i·17-s − 0.688·19-s + (−0.223 + 0.447i)20-s + 0.639i·22-s + (−0.600 − 0.800i)25-s + 1.37·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07962 + 0.667246i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07962 + 0.667246i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| 7 | \( 1 - iT \) |
| good | 2 | \( 1 + iT - 2T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 7iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 11T + 41T^{2} \) |
| 43 | \( 1 - 5iT - 43T^{2} \) |
| 47 | \( 1 - 9iT - 47T^{2} \) |
| 53 | \( 1 - 3iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 7iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 15iT - 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 - iT - 83T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−10.37809620991931755144409460468, −9.664682070354731271746202804498, −8.521907152001512226299167665749, −7.65043015072328358803321029365, −6.60659165941523676149445098861, −6.28090574662561220900071170407, −4.62404308084751685820420218927, −3.68216563265749536040987796800, −2.64689265799565623510362069605, −1.81749106988195979122706567054,
0.55467869404558245472692614666, 2.38140584795928833866537721242, 3.54884153493614247323759448890, 5.10881718692297764682261251742, 5.31211718995066015877005594865, 6.57024596159610698809319332991, 7.53634461850180133226083988418, 8.066231209283726939295948682156, 8.662346305912895277198102486764, 10.03489633630654000041739626521
