| L(s) = 1 | + 5-s + 4·11-s + 2·13-s − 2·17-s − 4·19-s + 8·23-s + 25-s − 2·29-s + 8·31-s − 37-s − 10·41-s − 12·43-s − 7·49-s + 6·53-s + 4·55-s + 4·59-s + 10·61-s + 2·65-s + 4·67-s − 8·71-s − 6·73-s − 8·79-s − 4·83-s − 2·85-s − 10·89-s − 4·95-s + 10·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.20·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 0.371·29-s + 1.43·31-s − 0.164·37-s − 1.56·41-s − 1.82·43-s − 49-s + 0.824·53-s + 0.539·55-s + 0.520·59-s + 1.28·61-s + 0.248·65-s + 0.488·67-s − 0.949·71-s − 0.702·73-s − 0.900·79-s − 0.439·83-s − 0.216·85-s − 1.05·89-s − 0.410·95-s + 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.85225947207323, −13.40311105825242, −13.05674443950432, −12.59192630331020, −11.85551365380521, −11.41114929036452, −11.22662601317772, −10.35785033994486, −10.07095816063581, −9.541933211579415, −8.793185871584083, −8.627789792710881, −8.235128558110644, −7.175044765936965, −6.825348904400384, −6.486756888919479, −5.938952515841875, −5.207227292590766, −4.736473858371124, −4.153852287974186, −3.510286462281239, −2.993347692516274, −2.218638961122952, −1.524643587932788, −1.059691171086559, 0,
1.059691171086559, 1.524643587932788, 2.218638961122952, 2.993347692516274, 3.510286462281239, 4.153852287974186, 4.736473858371124, 5.207227292590766, 5.938952515841875, 6.486756888919479, 6.825348904400384, 7.175044765936965, 8.235128558110644, 8.627789792710881, 8.793185871584083, 9.541933211579415, 10.07095816063581, 10.35785033994486, 11.22662601317772, 11.41114929036452, 11.85551365380521, 12.59192630331020, 13.05674443950432, 13.40311105825242, 13.85225947207323
