| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 11-s − 2·13-s − 15-s − 6·17-s − 8·19-s − 21-s − 6·23-s + 25-s − 27-s − 6·29-s + 2·31-s − 33-s + 35-s − 2·37-s + 2·39-s − 8·43-s + 45-s − 12·47-s + 49-s + 6·51-s − 6·53-s + 55-s + 8·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.258·15-s − 1.45·17-s − 1.83·19-s − 0.218·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.359·31-s − 0.174·33-s + 0.169·35-s − 0.328·37-s + 0.320·39-s − 1.21·43-s + 0.149·45-s − 1.75·47-s + 1/7·49-s + 0.840·51-s − 0.824·53-s + 0.134·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−14.65353225727698, −14.06404684770524, −13.49021499407672, −13.07044167978055, −12.61117525088943, −12.08854208614333, −11.46583545251654, −11.17300927230456, −10.52286261087474, −10.21077021695950, −9.527702268901852, −9.080222866617062, −8.444273400545872, −8.031144100797293, −7.320022053553899, −6.612299281867213, −6.388069038822815, −5.850485487052829, −5.095312222459352, −4.544185629379153, −4.241914514724398, −3.428560623159927, −2.515580358130391, −1.859585561681632, −1.580399133914402, 0, 0,
1.580399133914402, 1.859585561681632, 2.515580358130391, 3.428560623159927, 4.241914514724398, 4.544185629379153, 5.095312222459352, 5.850485487052829, 6.388069038822815, 6.612299281867213, 7.320022053553899, 8.031144100797293, 8.444273400545872, 9.080222866617062, 9.527702268901852, 10.21077021695950, 10.52286261087474, 11.17300927230456, 11.46583545251654, 12.08854208614333, 12.61117525088943, 13.07044167978055, 13.49021499407672, 14.06404684770524, 14.65353225727698
