| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 3·11-s + 4·13-s + 15-s − 2·17-s + 7·19-s + 21-s + 23-s + 25-s − 27-s − 4·29-s − 2·31-s − 3·33-s + 35-s + 4·37-s − 4·39-s + 9·41-s − 45-s + 3·47-s + 49-s + 2·51-s − 11·53-s − 3·55-s − 7·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.904·11-s + 1.10·13-s + 0.258·15-s − 0.485·17-s + 1.60·19-s + 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.742·29-s − 0.359·31-s − 0.522·33-s + 0.169·35-s + 0.657·37-s − 0.640·39-s + 1.40·41-s − 0.149·45-s + 0.437·47-s + 1/7·49-s + 0.280·51-s − 1.51·53-s − 0.404·55-s − 0.927·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.766846641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.766846641\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.29667109064263, −12.63837379682399, −12.48786924799663, −11.75828030707659, −11.28384366640079, −11.18685016096782, −10.61051359631517, −9.890902794209175, −9.418081422115764, −9.099021662347325, −8.575264203724700, −7.823589617332533, −7.441521365304090, −6.959175111878364, −6.379508741803730, −5.843847533853058, −5.630810291450564, −4.665216436492136, −4.361991947629906, −3.674015441688571, −3.287909043125922, −2.630773060452401, −1.579025881448420, −1.206412508850247, −0.4478538459043157,
0.4478538459043157, 1.206412508850247, 1.579025881448420, 2.630773060452401, 3.287909043125922, 3.674015441688571, 4.361991947629906, 4.665216436492136, 5.630810291450564, 5.843847533853058, 6.379508741803730, 6.959175111878364, 7.441521365304090, 7.823589617332533, 8.575264203724700, 9.099021662347325, 9.418081422115764, 9.890902794209175, 10.61051359631517, 11.18685016096782, 11.28384366640079, 11.75828030707659, 12.48786924799663, 12.63837379682399, 13.29667109064263
