| L(s) = 1 | − 2·4-s + 5-s − 3·9-s + 2·11-s − 6·13-s + 4·16-s − 5·17-s − 2·19-s − 2·20-s + 3·23-s + 25-s − 6·29-s − 4·31-s + 6·36-s − 7·37-s − 12·43-s − 4·44-s − 3·45-s − 6·47-s − 7·49-s + 12·52-s + 3·53-s + 2·55-s − 10·59-s − 2·61-s − 8·64-s − 6·65-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 9-s + 0.603·11-s − 1.66·13-s + 16-s − 1.21·17-s − 0.458·19-s − 0.447·20-s + 0.625·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s + 36-s − 1.15·37-s − 1.82·43-s − 0.603·44-s − 0.447·45-s − 0.875·47-s − 49-s + 1.66·52-s + 0.412·53-s + 0.269·55-s − 1.30·59-s − 0.256·61-s − 64-s − 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 6163 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.27552781508003, −14.98932916748937, −14.58704666492583, −14.03840648285451, −13.58376503343447, −13.01936042449564, −12.56036854604158, −11.98784821775843, −11.35605370666658, −10.85507119634175, −10.11853309538124, −9.637128962142948, −9.046503153679779, −8.833413332117972, −8.133306811460132, −7.445039264233371, −6.772583990290189, −6.227387334947741, −5.420396268260745, −4.978961448052874, −4.546193061327092, −3.634278025830638, −3.070862559957956, −2.196657573311608, −1.544688076191302, 0, 0,
1.544688076191302, 2.196657573311608, 3.070862559957956, 3.634278025830638, 4.546193061327092, 4.978961448052874, 5.420396268260745, 6.227387334947741, 6.772583990290189, 7.445039264233371, 8.133306811460132, 8.833413332117972, 9.046503153679779, 9.637128962142948, 10.11853309538124, 10.85507119634175, 11.35605370666658, 11.98784821775843, 12.56036854604158, 13.01936042449564, 13.58376503343447, 14.03840648285451, 14.58704666492583, 14.98932916748937, 15.27552781508003
