| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 3·9-s + 3·11-s − 14-s + 16-s − 8·17-s + 3·18-s + 7·19-s − 3·22-s − 23-s + 28-s + 7·29-s − 10·31-s − 32-s + 8·34-s − 3·36-s − 4·37-s − 7·38-s − 11·41-s + 5·43-s + 3·44-s + 46-s + 10·47-s − 6·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 9-s + 0.904·11-s − 0.267·14-s + 1/4·16-s − 1.94·17-s + 0.707·18-s + 1.60·19-s − 0.639·22-s − 0.208·23-s + 0.188·28-s + 1.29·29-s − 1.79·31-s − 0.176·32-s + 1.37·34-s − 1/2·36-s − 0.657·37-s − 1.13·38-s − 1.71·41-s + 0.762·43-s + 0.452·44-s + 0.147·46-s + 1.45·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6881199029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6881199029\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 11 T + p T^{2} \) | 1.41.l |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.18276676008002, −12.34294313332438, −12.06040590814222, −11.53338305845293, −11.29744070951406, −10.66824890532791, −10.45521670546471, −9.545246528089542, −9.156137416517875, −9.011747383674788, −8.336967575889239, −8.017390480156552, −7.306764622960461, −6.858172791074515, −6.470725504457526, −5.898500781656436, −5.207764210854523, −4.952926464426048, −4.027058428849680, −3.653665886717739, −2.873357655078732, −2.448710414265921, −1.677177632522227, −1.231476673853204, −0.2681144528357372,
0.2681144528357372, 1.231476673853204, 1.677177632522227, 2.448710414265921, 2.873357655078732, 3.653665886717739, 4.027058428849680, 4.952926464426048, 5.207764210854523, 5.898500781656436, 6.470725504457526, 6.858172791074515, 7.306764622960461, 8.017390480156552, 8.336967575889239, 9.011747383674788, 9.156137416517875, 9.545246528089542, 10.45521670546471, 10.66824890532791, 11.29744070951406, 11.53338305845293, 12.06040590814222, 12.34294313332438, 13.18276676008002
