L(s) = 1 | − 2·4-s + 5-s + 7-s + 4·13-s + 4·16-s + 2·17-s + 6·19-s − 2·20-s + 5·23-s − 4·25-s − 2·28-s + 10·29-s + 31-s + 35-s − 5·37-s − 2·41-s + 8·43-s − 8·47-s + 49-s − 8·52-s + 6·53-s − 3·59-s + 2·61-s − 8·64-s + 4·65-s − 3·67-s − 4·68-s + ⋯ |
L(s) = 1 | − 4-s + 0.447·5-s + 0.377·7-s + 1.10·13-s + 16-s + 0.485·17-s + 1.37·19-s − 0.447·20-s + 1.04·23-s − 4/5·25-s − 0.377·28-s + 1.85·29-s + 0.179·31-s + 0.169·35-s − 0.821·37-s − 0.312·41-s + 1.21·43-s − 1.16·47-s + 1/7·49-s − 1.10·52-s + 0.824·53-s − 0.390·59-s + 0.256·61-s − 64-s + 0.496·65-s − 0.366·67-s − 0.485·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.228041755\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.228041755\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.021783366018976031129107359000, −7.27725369419951075743399797902, −6.37025131836259169039920590021, −5.63513266845232393097119473069, −5.11970427972081013706150941332, −4.39032892467296438194436073318, −3.53296523383018775624550066684, −2.88258320316588856745096350068, −1.49123027748230791276986309146, −0.849034416833024727601479219042,
0.849034416833024727601479219042, 1.49123027748230791276986309146, 2.88258320316588856745096350068, 3.53296523383018775624550066684, 4.39032892467296438194436073318, 5.11970427972081013706150941332, 5.63513266845232393097119473069, 6.37025131836259169039920590021, 7.27725369419951075743399797902, 8.021783366018976031129107359000