| L(s) = 1 | + 1.62·3-s + 4.74·7-s − 0.367·9-s + 4.48·11-s + 0.843·13-s + 5.52·17-s + 19-s + 7.69·21-s − 0.779·23-s − 5.46·27-s − 10.6·29-s − 8.65·31-s + 7.26·33-s + 1.62·37-s + 1.36·39-s + 4.73·41-s − 9.67·43-s + 3.18·47-s + 15.5·49-s + 8.96·51-s − 6.17·53-s + 1.62·57-s + 11.6·59-s + 6.48·61-s − 1.74·63-s − 14.8·67-s − 1.26·69-s + ⋯ |
| L(s) = 1 | + 0.936·3-s + 1.79·7-s − 0.122·9-s + 1.35·11-s + 0.233·13-s + 1.33·17-s + 0.229·19-s + 1.67·21-s − 0.162·23-s − 1.05·27-s − 1.97·29-s − 1.55·31-s + 1.26·33-s + 0.266·37-s + 0.219·39-s + 0.739·41-s − 1.47·43-s + 0.464·47-s + 2.21·49-s + 1.25·51-s − 0.848·53-s + 0.214·57-s + 1.52·59-s + 0.829·61-s − 0.219·63-s − 1.81·67-s − 0.152·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.167049621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.167049621\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.62T + 3T^{2} \) |
| 7 | \( 1 - 4.74T + 7T^{2} \) |
| 11 | \( 1 - 4.48T + 11T^{2} \) |
| 13 | \( 1 - 0.843T + 13T^{2} \) |
| 17 | \( 1 - 5.52T + 17T^{2} \) |
| 23 | \( 1 + 0.779T + 23T^{2} \) |
| 29 | \( 1 + 10.6T + 29T^{2} \) |
| 31 | \( 1 + 8.65T + 31T^{2} \) |
| 37 | \( 1 - 1.62T + 37T^{2} \) |
| 41 | \( 1 - 4.73T + 41T^{2} \) |
| 43 | \( 1 + 9.67T + 43T^{2} \) |
| 47 | \( 1 - 3.18T + 47T^{2} \) |
| 53 | \( 1 + 6.17T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 6.48T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 - 0.303T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 0.779T + 83T^{2} \) |
| 89 | \( 1 + 5.69T + 89T^{2} \) |
| 97 | \( 1 + 6.17T + 97T^{2} \) |
| 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
−9.101949810872840805689289415059, −8.416412937564250017124565872062, −7.77075392860783843370917816042, −7.20487813847215237360647381153, −5.83556795931249361115846191233, −5.22065274786073324745615362808, −4.02293047528074274878704118832, −3.47170033987673154571460218050, −2.06490522029375114366641500384, −1.36604169372458639386397768431,
1.36604169372458639386397768431, 2.06490522029375114366641500384, 3.47170033987673154571460218050, 4.02293047528074274878704118832, 5.22065274786073324745615362808, 5.83556795931249361115846191233, 7.20487813847215237360647381153, 7.77075392860783843370917816042, 8.416412937564250017124565872062, 9.101949810872840805689289415059
