L(s) = 1 | − 1.61i·2-s + 1.18i·3-s − 0.621·4-s + (0.326 − 2.21i)5-s + 1.92·6-s + 2.23i·7-s − 2.23i·8-s + 1.58·9-s + (−3.58 − 0.528i)10-s + 5.64·11-s − 0.738i·12-s − 4.70i·13-s + 3.61·14-s + (2.62 + 0.387i)15-s − 4.85·16-s − 0.785i·17-s + ⋯ |
L(s) = 1 | − 1.14i·2-s + 0.686i·3-s − 0.310·4-s + (0.145 − 0.989i)5-s + 0.785·6-s + 0.843i·7-s − 0.789i·8-s + 0.529·9-s + (−1.13 − 0.167i)10-s + 1.70·11-s − 0.213i·12-s − 1.30i·13-s + 0.965·14-s + (0.678 + 0.100i)15-s − 1.21·16-s − 0.190i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.145 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.145 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.268370911\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.268370911\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.326 + 2.21i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 1.61iT - 2T^{2} \) |
| 3 | \( 1 - 1.18iT - 3T^{2} \) |
| 7 | \( 1 - 2.23iT - 7T^{2} \) |
| 11 | \( 1 - 5.64T + 11T^{2} \) |
| 13 | \( 1 + 4.70iT - 13T^{2} \) |
| 17 | \( 1 + 0.785iT - 17T^{2} \) |
| 23 | \( 1 - 5.13iT - 23T^{2} \) |
| 29 | \( 1 + 3.03T + 29T^{2} \) |
| 31 | \( 1 - 8.10T + 31T^{2} \) |
| 37 | \( 1 + 0.985iT - 37T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 - 1.52iT - 43T^{2} \) |
| 47 | \( 1 - 0.960iT - 47T^{2} \) |
| 53 | \( 1 + 4.41iT - 53T^{2} \) |
| 59 | \( 1 + 9.87T + 59T^{2} \) |
| 61 | \( 1 - 2.09T + 61T^{2} \) |
| 67 | \( 1 + 3.24iT - 67T^{2} \) |
| 71 | \( 1 - 7.17T + 71T^{2} \) |
| 73 | \( 1 + 15.1iT - 73T^{2} \) |
| 79 | \( 1 + 1.33T + 79T^{2} \) |
| 83 | \( 1 - 7.52iT - 83T^{2} \) |
| 89 | \( 1 + 3.40T + 89T^{2} \) |
| 97 | \( 1 + 12.7iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.482416415487784227091270118729, −8.696497349336638875094725668257, −7.65900465524409435838983551978, −6.50199387269277022512904013599, −5.62031509327840300344521413834, −4.69014436983183174940767886330, −3.90812101491879165658445014218, −3.13681431102054792091063986356, −1.83740962054823329304392514410, −0.974758034625289828719661350447,
1.36436372298983544483224765026, 2.38672465169715322564918254894, 3.91664535445974206052197333220, 4.50471690820731719208162655779, 6.07880456981466372944516698509, 6.57803489120295080028075703598, 6.92876989871200213666897118890, 7.50537924653085787213158807079, 8.434808376968541827020064698850, 9.327767431192157544346914598230