| L(s) = 1 | − 2-s + 4-s − 3·5-s − 8-s + 3·10-s + 6·11-s + 13-s + 16-s + 3·17-s − 2·19-s − 3·20-s − 6·22-s + 6·23-s + 4·25-s − 26-s + 9·29-s + 10·31-s − 32-s − 3·34-s − 7·37-s + 2·38-s + 3·40-s + 6·41-s − 4·43-s + 6·44-s − 6·46-s + 6·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.353·8-s + 0.948·10-s + 1.80·11-s + 0.277·13-s + 1/4·16-s + 0.727·17-s − 0.458·19-s − 0.670·20-s − 1.27·22-s + 1.25·23-s + 4/5·25-s − 0.196·26-s + 1.67·29-s + 1.79·31-s − 0.176·32-s − 0.514·34-s − 1.15·37-s + 0.324·38-s + 0.474·40-s + 0.937·41-s − 0.609·43-s + 0.904·44-s − 0.884·46-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.403335344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.403335344\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 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
−7.895626160485635645639447831647, −7.22491201169300574489774379035, −6.62122875679369166671317533937, −6.09659361780366225780678499454, −4.84353982835863770665530893587, −4.20858234757110523931060842705, −3.48654985780300195709992925545, −2.78234548155523523158272552207, −1.35995230440770390982176049155, −0.75014658543720990394544395207,
0.75014658543720990394544395207, 1.35995230440770390982176049155, 2.78234548155523523158272552207, 3.48654985780300195709992925545, 4.20858234757110523931060842705, 4.84353982835863770665530893587, 6.09659361780366225780678499454, 6.62122875679369166671317533937, 7.22491201169300574489774379035, 7.895626160485635645639447831647
