| L(s) = 1 | − 3-s + 2.92·5-s + 0.699·7-s + 9-s − 3.68·11-s − 1.47·13-s − 2.92·15-s − 1.28·17-s + 0.894·19-s − 0.699·21-s + 2.74·23-s + 3.54·25-s − 27-s − 5.50·29-s + 0.875·31-s + 3.68·33-s + 2.04·35-s − 8.06·37-s + 1.47·39-s − 9.13·41-s + 2.51·43-s + 2.92·45-s + 12.4·47-s − 6.51·49-s + 1.28·51-s + 4.68·53-s − 10.7·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.30·5-s + 0.264·7-s + 0.333·9-s − 1.11·11-s − 0.407·13-s − 0.754·15-s − 0.310·17-s + 0.205·19-s − 0.152·21-s + 0.571·23-s + 0.708·25-s − 0.192·27-s − 1.02·29-s + 0.157·31-s + 0.640·33-s + 0.345·35-s − 1.32·37-s + 0.235·39-s − 1.42·41-s + 0.383·43-s + 0.435·45-s + 1.82·47-s − 0.930·49-s + 0.179·51-s + 0.643·53-s − 1.45·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 503 | \( 1 - T \) |
| good | 5 | \( 1 - 2.92T + 5T^{2} \) |
| 7 | \( 1 - 0.699T + 7T^{2} \) |
| 11 | \( 1 + 3.68T + 11T^{2} \) |
| 13 | \( 1 + 1.47T + 13T^{2} \) |
| 17 | \( 1 + 1.28T + 17T^{2} \) |
| 19 | \( 1 - 0.894T + 19T^{2} \) |
| 23 | \( 1 - 2.74T + 23T^{2} \) |
| 29 | \( 1 + 5.50T + 29T^{2} \) |
| 31 | \( 1 - 0.875T + 31T^{2} \) |
| 37 | \( 1 + 8.06T + 37T^{2} \) |
| 41 | \( 1 + 9.13T + 41T^{2} \) |
| 43 | \( 1 - 2.51T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 4.68T + 53T^{2} \) |
| 59 | \( 1 - 0.385T + 59T^{2} \) |
| 61 | \( 1 - 6.91T + 61T^{2} \) |
| 67 | \( 1 + 9.28T + 67T^{2} \) |
| 71 | \( 1 + 7.71T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 0.468T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 - 8.59T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−7.50234277907931182664509007203, −7.04247500396334549078216480957, −6.15510419879035441841710217742, −5.41744053454246550934241694577, −5.20518415874524991091338009179, −4.23509091994461176952951060683, −3.03232299280317820199990690774, −2.21606274282366568081043890306, −1.43635318635286779898899064374, 0,
1.43635318635286779898899064374, 2.21606274282366568081043890306, 3.03232299280317820199990690774, 4.23509091994461176952951060683, 5.20518415874524991091338009179, 5.41744053454246550934241694577, 6.15510419879035441841710217742, 7.04247500396334549078216480957, 7.50234277907931182664509007203
