| L(s) = 1 | − 3·5-s + 7-s − 3·9-s + 6·11-s − 13-s + 4·17-s − 5·19-s − 3·23-s + 4·25-s − 5·29-s + 3·31-s − 3·35-s − 4·37-s − 6·41-s + 43-s + 9·45-s − 7·47-s + 49-s − 9·53-s − 18·55-s − 8·59-s − 10·61-s − 3·63-s + 3·65-s + 6·67-s + 8·71-s − 13·73-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s − 9-s + 1.80·11-s − 0.277·13-s + 0.970·17-s − 1.14·19-s − 0.625·23-s + 4/5·25-s − 0.928·29-s + 0.538·31-s − 0.507·35-s − 0.657·37-s − 0.937·41-s + 0.152·43-s + 1.34·45-s − 1.02·47-s + 1/7·49-s − 1.23·53-s − 2.42·55-s − 1.04·59-s − 1.28·61-s − 0.377·63-s + 0.372·65-s + 0.733·67-s + 0.949·71-s − 1.52·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−8.920719672465111957084037828236, −8.306653899443735547493881784402, −7.65856427227425769928738953272, −6.69324481208496363874135514467, −5.91107970337180073579003850586, −4.71772169836810783436044308640, −3.91316951599567757387300319071, −3.20361346457630305584783093767, −1.62321334030766970947302300597, 0,
1.62321334030766970947302300597, 3.20361346457630305584783093767, 3.91316951599567757387300319071, 4.71772169836810783436044308640, 5.91107970337180073579003850586, 6.69324481208496363874135514467, 7.65856427227425769928738953272, 8.306653899443735547493881784402, 8.920719672465111957084037828236
