| L(s) = 1 | − 5·3-s + 12·5-s − 11·7-s − 2·9-s − 54·11-s − 11·13-s − 60·15-s − 93·17-s + 19·19-s + 55·21-s − 183·23-s + 19·25-s + 145·27-s + 249·29-s − 56·31-s + 270·33-s − 132·35-s + 250·37-s + 55·39-s + 240·41-s − 196·43-s − 24·45-s + 168·47-s − 222·49-s + 465·51-s − 435·53-s − 648·55-s + ⋯ |
| L(s) = 1 | − 0.962·3-s + 1.07·5-s − 0.593·7-s − 0.0740·9-s − 1.48·11-s − 0.234·13-s − 1.03·15-s − 1.32·17-s + 0.229·19-s + 0.571·21-s − 1.65·23-s + 0.151·25-s + 1.03·27-s + 1.59·29-s − 0.324·31-s + 1.42·33-s − 0.637·35-s + 1.11·37-s + 0.225·39-s + 0.914·41-s − 0.695·43-s − 0.0795·45-s + 0.521·47-s − 0.647·49-s + 1.27·51-s − 1.12·53-s − 1.58·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7744765165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7744765165\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - p T \) |
| good | 3 | \( 1 + 5 T + p^{3} T^{2} \) |
| 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 7 | \( 1 + 11 T + p^{3} T^{2} \) |
| 11 | \( 1 + 54 T + p^{3} T^{2} \) |
| 13 | \( 1 + 11 T + p^{3} T^{2} \) |
| 17 | \( 1 + 93 T + p^{3} T^{2} \) |
| 23 | \( 1 + 183 T + p^{3} T^{2} \) |
| 29 | \( 1 - 249 T + p^{3} T^{2} \) |
| 31 | \( 1 + 56 T + p^{3} T^{2} \) |
| 37 | \( 1 - 250 T + p^{3} T^{2} \) |
| 41 | \( 1 - 240 T + p^{3} T^{2} \) |
| 43 | \( 1 + 196 T + p^{3} T^{2} \) |
| 47 | \( 1 - 168 T + p^{3} T^{2} \) |
| 53 | \( 1 + 435 T + p^{3} T^{2} \) |
| 59 | \( 1 - 195 T + p^{3} T^{2} \) |
| 61 | \( 1 - 358 T + p^{3} T^{2} \) |
| 67 | \( 1 + 961 T + p^{3} T^{2} \) |
| 71 | \( 1 - 246 T + p^{3} T^{2} \) |
| 73 | \( 1 - 353 T + p^{3} T^{2} \) |
| 79 | \( 1 - 34 T + p^{3} T^{2} \) |
| 83 | \( 1 - 234 T + p^{3} T^{2} \) |
| 89 | \( 1 + 168 T + p^{3} T^{2} \) |
| 97 | \( 1 - 758 T + p^{3} T^{2} \) |
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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
−9.634813026632343123838967073005, −8.582327758914048010612421239857, −7.67746222609057222332885786437, −6.44876638070725086725702022112, −6.09203399512015736597205127711, −5.27644191597752663308034316409, −4.47194618027410365365709873978, −2.87214808568039520043975223582, −2.10089315443453764424415898369, −0.44365427583020666769293067584,
0.44365427583020666769293067584, 2.10089315443453764424415898369, 2.87214808568039520043975223582, 4.47194618027410365365709873978, 5.27644191597752663308034316409, 6.09203399512015736597205127711, 6.44876638070725086725702022112, 7.67746222609057222332885786437, 8.582327758914048010612421239857, 9.634813026632343123838967073005
