| L(s) = 1 | − 3-s + 2·7-s + 9-s − 4·11-s − 2·13-s + 2·17-s − 8·19-s − 2·21-s + 4·23-s − 27-s + 6·31-s + 4·33-s + 2·37-s + 2·39-s + 6·41-s + 4·47-s − 3·49-s − 2·51-s + 8·57-s + 4·59-s + 14·61-s + 2·63-s + 4·67-s − 4·69-s − 12·71-s + 10·73-s − 8·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.485·17-s − 1.83·19-s − 0.436·21-s + 0.834·23-s − 0.192·27-s + 1.07·31-s + 0.696·33-s + 0.328·37-s + 0.320·39-s + 0.937·41-s + 0.583·47-s − 3/7·49-s − 0.280·51-s + 1.05·57-s + 0.520·59-s + 1.79·61-s + 0.251·63-s + 0.488·67-s − 0.481·69-s − 1.42·71-s + 1.17·73-s − 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−7.27427739257753081795590124388, −6.73736325834322400975552276771, −5.83485622447487237105598330171, −5.30922926633619698349818577199, −4.61836998817002689559950119242, −4.10164552347181392750315256357, −2.81220284738257297236567344088, −2.24861824058214226101578443816, −1.12672878009576529056910213451, 0,
1.12672878009576529056910213451, 2.24861824058214226101578443816, 2.81220284738257297236567344088, 4.10164552347181392750315256357, 4.61836998817002689559950119242, 5.30922926633619698349818577199, 5.83485622447487237105598330171, 6.73736325834322400975552276771, 7.27427739257753081795590124388
