| L(s) = 1 | + 3-s − 4·5-s − 2·7-s + 9-s − 4·15-s + 6·17-s − 4·19-s − 2·21-s + 6·23-s + 11·25-s + 27-s + 6·29-s + 8·35-s − 6·37-s + 10·41-s + 8·43-s − 4·45-s − 6·47-s − 3·49-s + 6·51-s + 12·53-s − 4·57-s − 8·59-s + 4·61-s − 2·63-s − 12·67-s + 6·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s − 0.755·7-s + 1/3·9-s − 1.03·15-s + 1.45·17-s − 0.917·19-s − 0.436·21-s + 1.25·23-s + 11/5·25-s + 0.192·27-s + 1.11·29-s + 1.35·35-s − 0.986·37-s + 1.56·41-s + 1.21·43-s − 0.596·45-s − 0.875·47-s − 3/7·49-s + 0.840·51-s + 1.64·53-s − 0.529·57-s − 1.04·59-s + 0.512·61-s − 0.251·63-s − 1.46·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.505990801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.505990801\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.48851784604566, −14.82908619007588, −14.66305186272688, −13.95033768889511, −13.16870743561289, −12.64876941146684, −12.29475422456976, −11.79800805634738, −11.08210250362253, −10.55546163337943, −10.03732630576263, −9.218364705612524, −8.718742908478063, −8.245692769394532, −7.554017771002983, −7.255994387357290, −6.594288701335768, −5.821478184604930, −4.922029509043111, −4.267303108328342, −3.787714534892289, −3.018225866861353, −2.794324872893504, −1.332661394119000, −0.5168954802565833,
0.5168954802565833, 1.332661394119000, 2.794324872893504, 3.018225866861353, 3.787714534892289, 4.267303108328342, 4.922029509043111, 5.821478184604930, 6.594288701335768, 7.255994387357290, 7.554017771002983, 8.245692769394532, 8.718742908478063, 9.218364705612524, 10.03732630576263, 10.55546163337943, 11.08210250362253, 11.79800805634738, 12.29475422456976, 12.64876941146684, 13.16870743561289, 13.95033768889511, 14.66305186272688, 14.82908619007588, 15.48851784604566
