| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s − 6·11-s − 12-s − 14-s + 16-s − 17-s + 18-s − 6·19-s + 21-s − 6·22-s − 4·23-s − 24-s − 27-s − 28-s − 4·29-s − 8·31-s + 32-s + 6·33-s − 34-s + 36-s − 4·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.80·11-s − 0.288·12-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 1.37·19-s + 0.218·21-s − 1.27·22-s − 0.834·23-s − 0.204·24-s − 0.192·27-s − 0.188·28-s − 0.742·29-s − 1.43·31-s + 0.176·32-s + 1.04·33-s − 0.171·34-s + 1/6·36-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8775865643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8775865643\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 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 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−15.70659443123148, −15.27999934375520, −14.91905356327782, −14.01859382886676, −13.49649279246394, −12.98962371394760, −12.61071960059299, −12.13865104238012, −11.27995141490389, −10.89493922621795, −10.29484983446283, −9.989621685591550, −8.958978677934267, −8.381520723045985, −7.557914530149268, −7.231000882542507, −6.273773559441614, −5.988842934094878, −5.108973576302682, −4.884836889567733, −3.868612892079188, −3.376250129786604, −2.287403435654144, −1.936200986795325, −0.3351632491443430,
0.3351632491443430, 1.936200986795325, 2.287403435654144, 3.376250129786604, 3.868612892079188, 4.884836889567733, 5.108973576302682, 5.988842934094878, 6.273773559441614, 7.231000882542507, 7.557914530149268, 8.381520723045985, 8.958978677934267, 9.989621685591550, 10.29484983446283, 10.89493922621795, 11.27995141490389, 12.13865104238012, 12.61071960059299, 12.98962371394760, 13.49649279246394, 14.01859382886676, 14.91905356327782, 15.27999934375520, 15.70659443123148
