| L(s) = 1 | + 3·3-s − 5·7-s + 6·9-s − 3·11-s − 4·13-s + 3·17-s − 4·19-s − 15·21-s + 9·27-s − 5·29-s + 5·31-s − 9·33-s − 3·37-s − 12·39-s − 2·41-s + 4·43-s − 2·47-s + 18·49-s + 9·51-s − 6·53-s − 12·57-s + 7·59-s + 15·61-s − 30·63-s + 6·67-s + 6·71-s + 10·73-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.88·7-s + 2·9-s − 0.904·11-s − 1.10·13-s + 0.727·17-s − 0.917·19-s − 3.27·21-s + 1.73·27-s − 0.928·29-s + 0.898·31-s − 1.56·33-s − 0.493·37-s − 1.92·39-s − 0.312·41-s + 0.609·43-s − 0.291·47-s + 18/7·49-s + 1.26·51-s − 0.824·53-s − 1.58·57-s + 0.911·59-s + 1.92·61-s − 3.77·63-s + 0.733·67-s + 0.712·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 83 | \( 1 - T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 - 15 T + p T^{2} \) | 1.61.ap |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−13.66872714392588, −13.23394780654551, −12.82540046351594, −12.50052420174133, −12.16525343358339, −11.19298819960725, −10.55484765592789, −10.02530214178554, −9.674326570278925, −9.557608779866849, −8.858904124691925, −8.232560355961596, −8.025857065952362, −7.293159239453971, −6.913660265487977, −6.522847557922015, −5.666310473467932, −5.204475566843659, −4.357875191488192, −3.815728758820165, −3.359125576743209, −2.854182050632825, −2.396578257471079, −1.990660226605682, −0.8012981468608993, 0,
0.8012981468608993, 1.990660226605682, 2.396578257471079, 2.854182050632825, 3.359125576743209, 3.815728758820165, 4.357875191488192, 5.204475566843659, 5.666310473467932, 6.522847557922015, 6.913660265487977, 7.293159239453971, 8.025857065952362, 8.232560355961596, 8.858904124691925, 9.557608779866849, 9.674326570278925, 10.02530214178554, 10.55484765592789, 11.19298819960725, 12.16525343358339, 12.50052420174133, 12.82540046351594, 13.23394780654551, 13.66872714392588
