| L(s) = 1 | + 3-s + 5-s + 7-s − 2·9-s + 2·13-s + 15-s − 17-s + 6·19-s + 21-s + 6·23-s − 4·25-s − 5·27-s + 5·31-s + 35-s − 8·37-s + 2·39-s − 6·41-s + 11·43-s − 2·45-s − 6·47-s + 49-s − 51-s − 9·53-s + 6·57-s − 4·59-s − 5·61-s − 2·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.554·13-s + 0.258·15-s − 0.242·17-s + 1.37·19-s + 0.218·21-s + 1.25·23-s − 4/5·25-s − 0.962·27-s + 0.898·31-s + 0.169·35-s − 1.31·37-s + 0.320·39-s − 0.937·41-s + 1.67·43-s − 0.298·45-s − 0.875·47-s + 1/7·49-s − 0.140·51-s − 1.23·53-s + 0.794·57-s − 0.520·59-s − 0.640·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.790813552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.790813552\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.05713802274276, −12.41879689233882, −12.04934804084234, −11.39065451769517, −11.14961870813416, −10.71473707824336, −9.970701596121985, −9.640716708507348, −9.055039949892041, −8.868583408564694, −8.103023528594661, −7.924968186100495, −7.333685436339262, −6.645040680510817, −6.322270633959279, −5.586292377071536, −5.241686614617886, −4.807376642552610, −4.010516227362382, −3.410749488573335, −3.078750272763998, −2.446030968596474, −1.832208318345849, −1.258107442411299, −0.5236021501202892,
0.5236021501202892, 1.258107442411299, 1.832208318345849, 2.446030968596474, 3.078750272763998, 3.410749488573335, 4.010516227362382, 4.807376642552610, 5.241686614617886, 5.586292377071536, 6.322270633959279, 6.645040680510817, 7.333685436339262, 7.924968186100495, 8.103023528594661, 8.868583408564694, 9.055039949892041, 9.640716708507348, 9.970701596121985, 10.71473707824336, 11.14961870813416, 11.39065451769517, 12.04934804084234, 12.41879689233882, 13.05713802274276
