L(s) = 1 | + (−1.58 + 1.58i)5-s + (3.23 + 3.23i)7-s − 4.57i·11-s + (−4.23 + 4.23i)13-s + (−1.74 + 1.74i)17-s + 2.47i·19-s + (2.82 + 2.82i)23-s − 5.00i·25-s − 5.99·29-s − 1.52·31-s − 10.2·35-s + (2.23 + 2.23i)37-s + 7.07i·41-s + (−2.47 + 2.47i)43-s + (−1.74 + 1.74i)47-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)5-s + (1.22 + 1.22i)7-s − 1.37i·11-s + (−1.17 + 1.17i)13-s + (−0.423 + 0.423i)17-s + 0.567i·19-s + (0.589 + 0.589i)23-s − 1.00i·25-s − 1.11·29-s − 0.274·31-s − 1.72·35-s + (0.367 + 0.367i)37-s + 1.10i·41-s + (−0.376 + 0.376i)43-s + (−0.254 + 0.254i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.641503 + 0.950742i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.641503 + 0.950742i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.58 - 1.58i)T \) |
good | 7 | \( 1 + (-3.23 - 3.23i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.57iT - 11T^{2} \) |
| 13 | \( 1 + (4.23 - 4.23i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.74 - 1.74i)T - 17iT^{2} \) |
| 19 | \( 1 - 2.47iT - 19T^{2} \) |
| 23 | \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 + 5.99T + 29T^{2} \) |
| 31 | \( 1 + 1.52T + 31T^{2} \) |
| 37 | \( 1 + (-2.23 - 2.23i)T + 37iT^{2} \) |
| 41 | \( 1 - 7.07iT - 41T^{2} \) |
| 43 | \( 1 + (2.47 - 2.47i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.74 - 1.74i)T - 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 - 1.08T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + (-1.52 - 1.52i)T + 67iT^{2} \) |
| 71 | \( 1 - 12.6iT - 71T^{2} \) |
| 73 | \( 1 + (0.527 - 0.527i)T - 73iT^{2} \) |
| 79 | \( 1 + 14.4iT - 79T^{2} \) |
| 83 | \( 1 + (-1.08 - 1.08i)T + 83iT^{2} \) |
| 89 | \( 1 + 0.746T + 89T^{2} \) |
| 97 | \( 1 + (-1 - i)T + 97iT^{2} \) |
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
−11.09078789246852231595408613829, −9.758835004467442066625579090269, −8.784386717431695932119487046724, −8.163435625348726627696081844746, −7.30656392608727575668667723919, −6.22422187239511441408728336906, −5.29118057547436272252294508995, −4.26151775814063534480678663285, −3.02405928160475353899795322665, −1.89874766529284240538796701137,
0.59204092447867449654813241752, 2.12819856696162460831603784942, 3.83430085014485089115926430086, 4.80070432049205321427114839097, 5.11445835870161938188251977500, 7.20564041878140050239033221548, 7.36432444245658328313826835758, 8.248284410477614197369578320908, 9.291887073990746617258380465087, 10.23165312147731553824670041443