| L(s) = 1 | + 5-s + 4·7-s − 4·11-s − 6·13-s − 6·17-s − 4·19-s + 25-s + 6·29-s + 4·35-s − 10·37-s − 2·41-s + 8·43-s + 12·47-s + 9·49-s − 14·53-s − 4·55-s + 4·59-s + 10·61-s − 6·65-s − 4·67-s − 8·71-s + 2·73-s − 16·77-s + 79-s − 4·83-s − 6·85-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s − 1.20·11-s − 1.66·13-s − 1.45·17-s − 0.917·19-s + 1/5·25-s + 1.11·29-s + 0.676·35-s − 1.64·37-s − 0.312·41-s + 1.21·43-s + 1.75·47-s + 9/7·49-s − 1.92·53-s − 0.539·55-s + 0.520·59-s + 1.28·61-s − 0.744·65-s − 0.488·67-s − 0.949·71-s + 0.234·73-s − 1.82·77-s + 0.112·79-s − 0.439·83-s − 0.650·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 227520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 79 | \( 1 - T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.10763692931964, −12.74410597731868, −12.16018870651525, −11.93190502085351, −11.08907502119853, −10.92811244157395, −10.34443959501514, −10.15260980009385, −9.373602665717253, −8.860259545262797, −8.466560716301661, −8.044401428621428, −7.412858583464404, −7.163949531324246, −6.529181391115572, −5.909745552217158, −5.209687110058598, −5.030291115803951, −4.471597412784422, −4.190518010143468, −3.087286532381989, −2.502145379018104, −2.132750342611407, −1.762128893322267, −0.7238932744952982, 0,
0.7238932744952982, 1.762128893322267, 2.132750342611407, 2.502145379018104, 3.087286532381989, 4.190518010143468, 4.471597412784422, 5.030291115803951, 5.209687110058598, 5.909745552217158, 6.529181391115572, 7.163949531324246, 7.412858583464404, 8.044401428621428, 8.466560716301661, 8.860259545262797, 9.373602665717253, 10.15260980009385, 10.34443959501514, 10.92811244157395, 11.08907502119853, 11.93190502085351, 12.16018870651525, 12.74410597731868, 13.10763692931964
