| L(s) = 1 | − 5-s − 4·11-s + 13-s + 6·17-s + 4·19-s − 4·23-s + 25-s + 10·29-s − 8·31-s + 6·37-s − 2·41-s − 4·43-s − 2·53-s + 4·55-s − 4·59-s − 10·61-s − 65-s + 4·67-s − 12·71-s − 10·73-s − 16·83-s − 6·85-s + 6·89-s − 4·95-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s + 0.277·13-s + 1.45·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.85·29-s − 1.43·31-s + 0.986·37-s − 0.312·41-s − 0.609·43-s − 0.274·53-s + 0.539·55-s − 0.520·59-s − 1.28·61-s − 0.124·65-s + 0.488·67-s − 1.42·71-s − 1.17·73-s − 1.75·83-s − 0.650·85-s + 0.635·89-s − 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.325951878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.325951878\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.91495353526281, −12.40161667826045, −11.90702521675606, −11.79561945039829, −10.91327732653950, −10.69954979633622, −10.10267584672188, −9.773975040475718, −9.276803609663443, −8.486775976918194, −8.230777763659413, −7.698036724682120, −7.407950048902471, −6.850388003579326, −6.055490555479480, −5.748495423312368, −5.215335158224427, −4.701717789686729, −4.155174446243092, −3.471257519819877, −2.984178785041681, −2.665223438329108, −1.655253284128733, −1.189593058031004, −0.3345940157640187,
0.3345940157640187, 1.189593058031004, 1.655253284128733, 2.665223438329108, 2.984178785041681, 3.471257519819877, 4.155174446243092, 4.701717789686729, 5.215335158224427, 5.748495423312368, 6.055490555479480, 6.850388003579326, 7.407950048902471, 7.698036724682120, 8.230777763659413, 8.486775976918194, 9.276803609663443, 9.773975040475718, 10.10267584672188, 10.69954979633622, 10.91327732653950, 11.79561945039829, 11.90702521675606, 12.40161667826045, 12.91495353526281
