| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s + 3·11-s − 4·13-s + 15-s − 5·19-s + 2·21-s + 25-s − 27-s − 3·29-s − 8·31-s − 3·33-s + 2·35-s + 2·37-s + 4·39-s − 3·41-s + 10·43-s − 45-s + 6·47-s − 3·49-s − 6·53-s − 3·55-s + 5·57-s − 15·59-s + 5·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.904·11-s − 1.10·13-s + 0.258·15-s − 1.14·19-s + 0.436·21-s + 1/5·25-s − 0.192·27-s − 0.557·29-s − 1.43·31-s − 0.522·33-s + 0.338·35-s + 0.328·37-s + 0.640·39-s − 0.468·41-s + 1.52·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s − 0.824·53-s − 0.404·55-s + 0.662·57-s − 1.95·59-s + 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 15 T + p T^{2} \) | 1.59.p |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.47385774812292, −13.98205007654015, −13.22867958703776, −12.73561899627803, −12.33956602234585, −12.10439371598135, −11.31854162215707, −10.88992574398115, −10.54404265987921, −9.725642743509861, −9.314414451764731, −9.065040365106745, −8.119736577656333, −7.687474912061749, −6.966350597772125, −6.748385000866691, −6.041248840749501, −5.603369304996376, −4.812845779424294, −4.319946749314948, −3.766621698423646, −3.181582597653516, −2.316352028336008, −1.720451891633977, −0.6723402195493266, 0,
0.6723402195493266, 1.720451891633977, 2.316352028336008, 3.181582597653516, 3.766621698423646, 4.319946749314948, 4.812845779424294, 5.603369304996376, 6.041248840749501, 6.748385000866691, 6.966350597772125, 7.687474912061749, 8.119736577656333, 9.065040365106745, 9.314414451764731, 9.725642743509861, 10.54404265987921, 10.88992574398115, 11.31854162215707, 12.10439371598135, 12.33956602234585, 12.73561899627803, 13.22867958703776, 13.98205007654015, 14.47385774812292
