L(s) = 1 | − 5-s + 7-s + 4·11-s − 2·13-s + 3·17-s + 8·19-s + 6·23-s − 4·25-s − 4·29-s − 6·31-s − 35-s − 3·37-s + 41-s − 11·43-s − 9·47-s + 49-s + 6·53-s − 4·55-s + 15·59-s + 4·61-s + 2·65-s + 8·67-s + 12·71-s + 6·73-s + 4·77-s + 79-s + 9·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s + 1.20·11-s − 0.554·13-s + 0.727·17-s + 1.83·19-s + 1.25·23-s − 4/5·25-s − 0.742·29-s − 1.07·31-s − 0.169·35-s − 0.493·37-s + 0.156·41-s − 1.67·43-s − 1.31·47-s + 1/7·49-s + 0.824·53-s − 0.539·55-s + 1.95·59-s + 0.512·61-s + 0.248·65-s + 0.977·67-s + 1.42·71-s + 0.702·73-s + 0.455·77-s + 0.112·79-s + 0.987·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.961476588\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.961476588\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.729485569358290065460316314758, −7.88452133072696678604809936942, −7.25251494070528755481669441144, −6.65813577807847277238815424142, −5.40801773340443895405634653905, −5.05971017349345746635432394949, −3.76204991759879263596774310983, −3.37423655942682476559491319761, −1.94425866003935808730361221120, −0.895372083176175841154288549329,
0.895372083176175841154288549329, 1.94425866003935808730361221120, 3.37423655942682476559491319761, 3.76204991759879263596774310983, 5.05971017349345746635432394949, 5.40801773340443895405634653905, 6.65813577807847277238815424142, 7.25251494070528755481669441144, 7.88452133072696678604809936942, 8.729485569358290065460316314758