| L(s) = 1 | + 3-s − 2·7-s + 9-s − 2·13-s + 8·17-s − 2·21-s + 23-s − 5·25-s + 27-s − 2·29-s + 4·31-s − 6·37-s − 2·39-s − 10·41-s + 6·43-s − 3·49-s + 8·51-s − 12·53-s − 4·59-s − 10·61-s − 2·63-s + 6·67-s + 69-s + 2·73-s − 5·75-s + 6·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.554·13-s + 1.94·17-s − 0.436·21-s + 0.208·23-s − 25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.986·37-s − 0.320·39-s − 1.56·41-s + 0.914·43-s − 3/7·49-s + 1.12·51-s − 1.64·53-s − 0.520·59-s − 1.28·61-s − 0.251·63-s + 0.733·67-s + 0.120·69-s + 0.234·73-s − 0.577·75-s + 0.675·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 199272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 199272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.27419626720178, −12.79848184987562, −12.42597982355693, −11.93323996022786, −11.65338407714834, −10.78832173671951, −10.37488408452877, −9.919343118610112, −9.502057045753533, −9.273201630868792, −8.436416508345738, −8.100420508600434, −7.486244624469352, −7.314663377531442, −6.464828983683244, −6.166795526645995, −5.501653024669449, −4.992930382529547, −4.469003894016630, −3.632424321206991, −3.310763871085168, −2.988058263189855, −2.097629686384629, −1.627991839801812, −0.8130823468539019, 0,
0.8130823468539019, 1.627991839801812, 2.097629686384629, 2.988058263189855, 3.310763871085168, 3.632424321206991, 4.469003894016630, 4.992930382529547, 5.501653024669449, 6.166795526645995, 6.464828983683244, 7.314663377531442, 7.486244624469352, 8.100420508600434, 8.436416508345738, 9.273201630868792, 9.502057045753533, 9.919343118610112, 10.37488408452877, 10.78832173671951, 11.65338407714834, 11.93323996022786, 12.42597982355693, 12.79848184987562, 13.27419626720178
