| L(s) = 1 | − 3·3-s + 4·7-s + 6·9-s + 3·13-s − 3·17-s − 5·19-s − 12·21-s − 7·23-s − 5·25-s − 9·27-s + 7·29-s − 4·31-s − 5·37-s − 9·39-s + 6·41-s − 2·43-s + 2·47-s + 9·49-s + 9·51-s + 53-s + 15·57-s − 2·59-s + 8·61-s + 24·63-s − 12·67-s + 21·69-s − 71-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.51·7-s + 2·9-s + 0.832·13-s − 0.727·17-s − 1.14·19-s − 2.61·21-s − 1.45·23-s − 25-s − 1.73·27-s + 1.29·29-s − 0.718·31-s − 0.821·37-s − 1.44·39-s + 0.937·41-s − 0.304·43-s + 0.291·47-s + 9/7·49-s + 1.26·51-s + 0.137·53-s + 1.98·57-s − 0.260·59-s + 1.02·61-s + 3.02·63-s − 1.46·67-s + 2.52·69-s − 0.118·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−8.205174103101839964623123933331, −7.40585307678501251492741626335, −6.49177047979490854052512324082, −5.95808888910984261709863888066, −5.27863549534731961075900677576, −4.42654752979487491573226444690, −4.04318522706247983721491253685, −2.12271179066542804694951246257, −1.33598343949561168216183034148, 0,
1.33598343949561168216183034148, 2.12271179066542804694951246257, 4.04318522706247983721491253685, 4.42654752979487491573226444690, 5.27863549534731961075900677576, 5.95808888910984261709863888066, 6.49177047979490854052512324082, 7.40585307678501251492741626335, 8.205174103101839964623123933331
