| L(s) = 1 | − 2-s − 4-s + 3·8-s − 2·11-s + 6·13-s − 16-s − 8·17-s − 19-s + 2·22-s − 6·23-s − 5·25-s − 6·26-s + 6·29-s − 5·32-s + 8·34-s − 2·37-s + 38-s − 2·41-s − 4·43-s + 2·44-s + 6·46-s + 2·47-s + 5·50-s − 6·52-s + 6·53-s − 6·58-s + 4·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 0.603·11-s + 1.66·13-s − 1/4·16-s − 1.94·17-s − 0.229·19-s + 0.426·22-s − 1.25·23-s − 25-s − 1.17·26-s + 1.11·29-s − 0.883·32-s + 1.37·34-s − 0.328·37-s + 0.162·38-s − 0.312·41-s − 0.609·43-s + 0.301·44-s + 0.884·46-s + 0.291·47-s + 0.707·50-s − 0.832·52-s + 0.824·53-s − 0.787·58-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7577506500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7577506500\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 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.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−8.146377640435571044349460550987, −7.20228910784419817052518066587, −6.47237508788768692091623584934, −5.82994999187726437210022046143, −4.94711103783663801276664879859, −4.16098249749107196218869175841, −3.71769837034904988951400042275, −2.40542862152543239693569966338, −1.64028916967236157381138626713, −0.48349128478257683481484162786,
0.48349128478257683481484162786, 1.64028916967236157381138626713, 2.40542862152543239693569966338, 3.71769837034904988951400042275, 4.16098249749107196218869175841, 4.94711103783663801276664879859, 5.82994999187726437210022046143, 6.47237508788768692091623584934, 7.20228910784419817052518066587, 8.146377640435571044349460550987
