| L(s) = 1 | + 27·3-s − 26·5-s − 1.05e3·7-s + 729·9-s − 6.41e3·11-s + 5.20e3·13-s − 702·15-s − 6.23e3·17-s − 4.14e4·19-s − 2.85e4·21-s + 2.94e4·23-s − 7.74e4·25-s + 1.96e4·27-s − 2.10e5·29-s − 1.85e5·31-s − 1.73e5·33-s + 2.74e4·35-s + 5.07e5·37-s + 1.40e5·39-s + 3.60e5·41-s − 6.20e5·43-s − 1.89e4·45-s + 8.47e5·47-s + 2.91e5·49-s − 1.68e5·51-s + 1.42e6·53-s + 1.66e5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.0930·5-s − 1.16·7-s + 1/3·9-s − 1.45·11-s + 0.657·13-s − 0.0537·15-s − 0.307·17-s − 1.38·19-s − 0.671·21-s + 0.504·23-s − 0.991·25-s + 0.192·27-s − 1.60·29-s − 1.11·31-s − 0.838·33-s + 0.108·35-s + 1.64·37-s + 0.379·39-s + 0.815·41-s − 1.18·43-s − 0.0310·45-s + 1.19·47-s + 0.354·49-s − 0.177·51-s + 1.31·53-s + 0.135·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{3} T \) |
| good | 5 | \( 1 + 26 T + p^{7} T^{2} \) |
| 7 | \( 1 + 1056 T + p^{7} T^{2} \) |
| 11 | \( 1 + 6412 T + p^{7} T^{2} \) |
| 13 | \( 1 - 5206 T + p^{7} T^{2} \) |
| 17 | \( 1 + 6238 T + p^{7} T^{2} \) |
| 19 | \( 1 + 41492 T + p^{7} T^{2} \) |
| 23 | \( 1 - 29432 T + p^{7} T^{2} \) |
| 29 | \( 1 + 210498 T + p^{7} T^{2} \) |
| 31 | \( 1 + 185240 T + p^{7} T^{2} \) |
| 37 | \( 1 - 507630 T + p^{7} T^{2} \) |
| 41 | \( 1 - 360042 T + p^{7} T^{2} \) |
| 43 | \( 1 + 620044 T + p^{7} T^{2} \) |
| 47 | \( 1 - 847680 T + p^{7} T^{2} \) |
| 53 | \( 1 - 1423750 T + p^{7} T^{2} \) |
| 59 | \( 1 - 2548724 T + p^{7} T^{2} \) |
| 61 | \( 1 + 706058 T + p^{7} T^{2} \) |
| 67 | \( 1 - 2418796 T + p^{7} T^{2} \) |
| 71 | \( 1 + 265976 T + p^{7} T^{2} \) |
| 73 | \( 1 + 5791238 T + p^{7} T^{2} \) |
| 79 | \( 1 + 2955688 T + p^{7} T^{2} \) |
| 83 | \( 1 + 3462932 T + p^{7} T^{2} \) |
| 89 | \( 1 + 2211126 T + p^{7} T^{2} \) |
| 97 | \( 1 + 15594814 T + p^{7} T^{2} \) |
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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
−13.13273670616632380300700129939, −13.03288315775664451280571003405, −11.05334461119430316061189230172, −9.896923050005125187814335985164, −8.681275451515645146454298564389, −7.36296398411236702394344515750, −5.85312353623453881465198245315, −3.87072238832342680346189485530, −2.42145820658498808701778317823, 0,
2.42145820658498808701778317823, 3.87072238832342680346189485530, 5.85312353623453881465198245315, 7.36296398411236702394344515750, 8.681275451515645146454298564389, 9.896923050005125187814335985164, 11.05334461119430316061189230172, 13.03288315775664451280571003405, 13.13273670616632380300700129939
