| L(s) = 1 | − 20·3-s + 74·5-s + 157·9-s + 124·11-s − 478·13-s − 1.48e3·15-s + 1.19e3·17-s − 3.04e3·19-s + 184·23-s + 2.35e3·25-s + 1.72e3·27-s − 3.28e3·29-s + 5.72e3·31-s − 2.48e3·33-s + 1.03e4·37-s + 9.56e3·39-s + 8.88e3·41-s − 9.18e3·43-s + 1.16e4·45-s − 2.36e4·47-s − 2.39e4·51-s + 1.16e4·53-s + 9.17e3·55-s + 6.08e4·57-s − 1.68e4·59-s + 1.84e4·61-s − 3.53e4·65-s + ⋯ |
| L(s) = 1 | − 1.28·3-s + 1.32·5-s + 0.646·9-s + 0.308·11-s − 0.784·13-s − 1.69·15-s + 1.00·17-s − 1.93·19-s + 0.0725·23-s + 0.752·25-s + 0.454·27-s − 0.724·29-s + 1.07·31-s − 0.396·33-s + 1.24·37-s + 1.00·39-s + 0.825·41-s − 0.757·43-s + 0.855·45-s − 1.56·47-s − 1.28·51-s + 0.571·53-s + 0.409·55-s + 2.48·57-s − 0.631·59-s + 0.635·61-s − 1.03·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 20 T + p^{5} T^{2} \) |
| 5 | \( 1 - 74 T + p^{5} T^{2} \) |
| 11 | \( 1 - 124 T + p^{5} T^{2} \) |
| 13 | \( 1 + 478 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1198 T + p^{5} T^{2} \) |
| 19 | \( 1 + 3044 T + p^{5} T^{2} \) |
| 23 | \( 1 - 8 p T + p^{5} T^{2} \) |
| 29 | \( 1 + 3282 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5728 T + p^{5} T^{2} \) |
| 37 | \( 1 - 10326 T + p^{5} T^{2} \) |
| 41 | \( 1 - 8886 T + p^{5} T^{2} \) |
| 43 | \( 1 + 9188 T + p^{5} T^{2} \) |
| 47 | \( 1 + 23664 T + p^{5} T^{2} \) |
| 53 | \( 1 - 11686 T + p^{5} T^{2} \) |
| 59 | \( 1 + 16876 T + p^{5} T^{2} \) |
| 61 | \( 1 - 18482 T + p^{5} T^{2} \) |
| 67 | \( 1 + 15532 T + p^{5} T^{2} \) |
| 71 | \( 1 + 31960 T + p^{5} T^{2} \) |
| 73 | \( 1 - 4886 T + p^{5} T^{2} \) |
| 79 | \( 1 - 44560 T + p^{5} T^{2} \) |
| 83 | \( 1 + 67364 T + p^{5} T^{2} \) |
| 89 | \( 1 + 71994 T + p^{5} T^{2} \) |
| 97 | \( 1 + 48866 T + p^{5} T^{2} \) |
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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
−10.11222346291505666033096583388, −9.455995787613354179364649354085, −8.177827496373352711994823752708, −6.76546897425176225451516603760, −6.10819826823687803207083366393, −5.39078064401797257772383004948, −4.39548302817365484310860293740, −2.57053165742938981066271684668, −1.35856208423527519470182892394, 0,
1.35856208423527519470182892394, 2.57053165742938981066271684668, 4.39548302817365484310860293740, 5.39078064401797257772383004948, 6.10819826823687803207083366393, 6.76546897425176225451516603760, 8.177827496373352711994823752708, 9.455995787613354179364649354085, 10.11222346291505666033096583388
