| L(s) = 1 | + i·5-s + 3.41i·7-s + 1.41·11-s + 6.24·13-s − 6.82i·17-s + 5.65i·19-s + 8.82·23-s − 25-s + 2i·29-s − 8.82i·31-s − 3.41·35-s − 7.41·37-s + 0.242i·41-s − 1.17i·43-s + 8·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s + 1.29i·7-s + 0.426·11-s + 1.73·13-s − 1.65i·17-s + 1.29i·19-s + 1.84·23-s − 0.200·25-s + 0.371i·29-s − 1.58i·31-s − 0.577·35-s − 1.21·37-s + 0.0378i·41-s − 0.178i·43-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.139053381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.139053381\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| good | 7 | \( 1 - 3.41iT - 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 6.24T + 13T^{2} \) |
| 17 | \( 1 + 6.82iT - 17T^{2} \) |
| 19 | \( 1 - 5.65iT - 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 8.82iT - 31T^{2} \) |
| 37 | \( 1 + 7.41T + 37T^{2} \) |
| 41 | \( 1 - 0.242iT - 41T^{2} \) |
| 43 | \( 1 + 1.17iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 1.17iT - 53T^{2} \) |
| 59 | \( 1 + 2.58T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 - 11.3iT - 67T^{2} \) |
| 71 | \( 1 - 2.34T + 71T^{2} \) |
| 73 | \( 1 + 4.82T + 73T^{2} \) |
| 79 | \( 1 - 10.4iT - 79T^{2} \) |
| 83 | \( 1 - 6.82T + 83T^{2} \) |
| 89 | \( 1 + 15.0iT - 89T^{2} \) |
| 97 | \( 1 + 6.48T + 97T^{2} \) |
| 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
−8.953598698468197280468288677326, −8.290638124688245956372963594313, −7.30290567341433790840767783625, −6.58142118011753634493508451321, −5.76113325791295245643264329391, −5.27492608265955654688357862095, −4.01074942333532984837396702067, −3.19665671374853269105855006809, −2.35518719012993762777389727016, −1.12875781980301287431240731933,
0.869287285908825775935277167964, 1.56359179388142762737386858394, 3.24316953126698294378336380890, 3.87407782867160857536919733055, 4.63489272315495026340548773387, 5.56669624344414198530172293115, 6.61881724718447907601107081381, 6.93584810286096691288251523641, 8.017542110984758294560632536516, 8.755545966607491779650745838600
