| L(s) = 1 | − 5-s + 4·7-s − 3·9-s + 11-s − 6·13-s − 6·17-s − 4·19-s + 4·23-s + 25-s + 2·29-s + 8·31-s − 4·35-s + 10·37-s + 10·41-s + 3·45-s + 4·47-s + 9·49-s + 10·53-s − 55-s + 4·59-s + 2·61-s − 12·63-s + 6·65-s + 8·67-s − 14·73-s + 4·77-s − 16·79-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 9-s + 0.301·11-s − 1.66·13-s − 1.45·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.676·35-s + 1.64·37-s + 1.56·41-s + 0.447·45-s + 0.583·47-s + 9/7·49-s + 1.37·53-s − 0.134·55-s + 0.520·59-s + 0.256·61-s − 1.51·63-s + 0.744·65-s + 0.977·67-s − 1.63·73-s + 0.455·77-s − 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.580859577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.580859577\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 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 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−8.603124223940541989763355707345, −7.86875493420056316464330358093, −7.23681139726552369590830113224, −6.37688806503460867410431993597, −5.43357459184150838511384967210, −4.49963301673034241264130696902, −4.38100491156023694629623840508, −2.68898784528613655503149585492, −2.26335656784139746959742713088, −0.72422113221940546069698321958,
0.72422113221940546069698321958, 2.26335656784139746959742713088, 2.68898784528613655503149585492, 4.38100491156023694629623840508, 4.49963301673034241264130696902, 5.43357459184150838511384967210, 6.37688806503460867410431993597, 7.23681139726552369590830113224, 7.86875493420056316464330358093, 8.603124223940541989763355707345
