| L(s) = 1 | + 3-s − 2·9-s + 5·11-s − 13-s + 3·17-s − 6·19-s + 6·23-s − 5·27-s + 9·29-s + 5·33-s + 6·37-s − 39-s − 8·41-s + 6·43-s + 3·47-s + 3·51-s − 12·53-s − 6·57-s + 8·59-s − 4·61-s − 4·67-s + 6·69-s + 8·71-s + 10·73-s − 3·79-s + 81-s + 12·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s + 1.50·11-s − 0.277·13-s + 0.727·17-s − 1.37·19-s + 1.25·23-s − 0.962·27-s + 1.67·29-s + 0.870·33-s + 0.986·37-s − 0.160·39-s − 1.24·41-s + 0.914·43-s + 0.437·47-s + 0.420·51-s − 1.64·53-s − 0.794·57-s + 1.04·59-s − 0.512·61-s − 0.488·67-s + 0.722·69-s + 0.949·71-s + 1.17·73-s − 0.337·79-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.427766195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.427766195\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.10874926469679, −13.62716593647438, −13.09840413167822, −12.37038847258644, −12.16606731529747, −11.55465156660090, −11.00608927220126, −10.59855466366265, −9.870612589587400, −9.327395749554864, −9.043741513506315, −8.358364142505813, −8.158345467803904, −7.371124171431495, −6.745404850383472, −6.327770196998383, −5.871705312994056, −4.949858560333946, −4.606237128903756, −3.793978955809463, −3.379450291057818, −2.694837601810739, −2.137958604605278, −1.283167661101994, −0.6186090706622158,
0.6186090706622158, 1.283167661101994, 2.137958604605278, 2.694837601810739, 3.379450291057818, 3.793978955809463, 4.606237128903756, 4.949858560333946, 5.871705312994056, 6.327770196998383, 6.745404850383472, 7.371124171431495, 8.158345467803904, 8.358364142505813, 9.043741513506315, 9.327395749554864, 9.870612589587400, 10.59855466366265, 11.00608927220126, 11.55465156660090, 12.16606731529747, 12.37038847258644, 13.09840413167822, 13.62716593647438, 14.10874926469679
