| L(s) = 1 | + 5-s − 7-s − 2·17-s + 2·19-s + 7·23-s − 4·25-s − 10·29-s + 7·31-s − 35-s − 9·37-s − 2·41-s + 4·43-s − 8·47-s + 49-s − 2·53-s + 15·59-s + 14·61-s + 3·67-s − 3·71-s − 10·73-s − 10·79-s − 2·85-s + 11·89-s + 2·95-s + 7·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.485·17-s + 0.458·19-s + 1.45·23-s − 4/5·25-s − 1.85·29-s + 1.25·31-s − 0.169·35-s − 1.47·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.274·53-s + 1.95·59-s + 1.79·61-s + 0.366·67-s − 0.356·71-s − 1.17·73-s − 1.12·79-s − 0.216·85-s + 1.16·89-s + 0.205·95-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 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
−14.65884701787377, −13.97958527862793, −13.39768046511305, −13.10820434122518, −12.74785215618790, −11.89374981947003, −11.50997820626868, −11.10294121465128, −10.30804066919671, −9.997862488261035, −9.444348051979910, −8.899584715867101, −8.506378295084898, −7.726043838658613, −7.177280763358931, −6.724395708285619, −6.165725165434914, −5.391914294623577, −5.203034675709805, −4.325943120442058, −3.672106236064741, −3.140521972138192, −2.378557664732667, −1.781052101195630, −0.9467989290149556, 0,
0.9467989290149556, 1.781052101195630, 2.378557664732667, 3.140521972138192, 3.672106236064741, 4.325943120442058, 5.203034675709805, 5.391914294623577, 6.165725165434914, 6.724395708285619, 7.177280763358931, 7.726043838658613, 8.506378295084898, 8.899584715867101, 9.444348051979910, 9.997862488261035, 10.30804066919671, 11.10294121465128, 11.50997820626868, 11.89374981947003, 12.74785215618790, 13.10820434122518, 13.39768046511305, 13.97958527862793, 14.65884701787377
