| L(s) = 1 | − 390·5-s + 64·7-s − 948·11-s − 5.09e3·13-s − 2.83e4·17-s + 8.62e3·19-s − 1.52e4·23-s + 7.39e4·25-s − 3.65e4·29-s + 2.76e5·31-s − 2.49e4·35-s + 2.68e5·37-s + 6.29e5·41-s − 6.85e5·43-s + 5.83e5·47-s − 8.19e5·49-s + 4.28e5·53-s + 3.69e5·55-s + 1.30e6·59-s + 3.00e5·61-s + 1.98e6·65-s + 5.07e5·67-s + 5.56e6·71-s + 1.36e6·73-s − 6.06e4·77-s + 6.91e6·79-s − 4.37e6·83-s + ⋯ |
| L(s) = 1 | − 1.39·5-s + 0.0705·7-s − 0.214·11-s − 0.643·13-s − 1.40·17-s + 0.288·19-s − 0.262·23-s + 0.946·25-s − 0.277·29-s + 1.66·31-s − 0.0984·35-s + 0.871·37-s + 1.42·41-s − 1.31·43-s + 0.819·47-s − 0.995·49-s + 0.394·53-s + 0.299·55-s + 0.828·59-s + 0.169·61-s + 0.897·65-s + 0.206·67-s + 1.84·71-s + 0.411·73-s − 0.0151·77-s + 1.57·79-s − 0.840·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.039546941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.039546941\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 78 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 64 T + p^{7} T^{2} \) |
| 11 | \( 1 + 948 T + p^{7} T^{2} \) |
| 13 | \( 1 + 5098 T + p^{7} T^{2} \) |
| 17 | \( 1 + 28386 T + p^{7} T^{2} \) |
| 19 | \( 1 - 8620 T + p^{7} T^{2} \) |
| 23 | \( 1 + 15288 T + p^{7} T^{2} \) |
| 29 | \( 1 + 36510 T + p^{7} T^{2} \) |
| 31 | \( 1 - 276808 T + p^{7} T^{2} \) |
| 37 | \( 1 - 268526 T + p^{7} T^{2} \) |
| 41 | \( 1 - 629718 T + p^{7} T^{2} \) |
| 43 | \( 1 + 685772 T + p^{7} T^{2} \) |
| 47 | \( 1 - 583296 T + p^{7} T^{2} \) |
| 53 | \( 1 - 428058 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1306380 T + p^{7} T^{2} \) |
| 61 | \( 1 - 300662 T + p^{7} T^{2} \) |
| 67 | \( 1 - 507244 T + p^{7} T^{2} \) |
| 71 | \( 1 - 5560632 T + p^{7} T^{2} \) |
| 73 | \( 1 - 1369082 T + p^{7} T^{2} \) |
| 79 | \( 1 - 6913720 T + p^{7} T^{2} \) |
| 83 | \( 1 + 4376748 T + p^{7} T^{2} \) |
| 89 | \( 1 - 8528310 T + p^{7} T^{2} \) |
| 97 | \( 1 + 8826814 T + p^{7} T^{2} \) |
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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
−11.69541019043434953352054423345, −10.99743182494695270390021804029, −9.700608685129881041339734081008, −8.430595937428673121836881544600, −7.63796430226373725074195221715, −6.55252849405598754213087188327, −4.87263005145203287059354009368, −3.93630846593330550932936996004, −2.50769877878182521761714632741, −0.56577320808709774363426222447,
0.56577320808709774363426222447, 2.50769877878182521761714632741, 3.93630846593330550932936996004, 4.87263005145203287059354009368, 6.55252849405598754213087188327, 7.63796430226373725074195221715, 8.430595937428673121836881544600, 9.700608685129881041339734081008, 10.99743182494695270390021804029, 11.69541019043434953352054423345
