| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 12-s + 13-s − 15-s + 16-s − 2·17-s − 18-s − 20-s − 6·23-s − 24-s + 25-s − 26-s + 27-s − 6·29-s + 30-s + 8·31-s − 32-s + 2·34-s + 36-s − 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.223·20-s − 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 1.11·29-s + 0.182·30-s + 1.43·31-s − 0.176·32-s + 0.342·34-s + 1/6·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.414730487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.414730487\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−15.64685373784225, −15.44634616244571, −14.64197294125435, −14.19738450521627, −13.55918406892001, −13.00486023850522, −12.31697985296928, −11.82658911958211, −11.22166062198454, −10.66481580843974, −10.04837299742061, −9.527098494299203, −8.887940894967668, −8.422696942539207, −7.807905915524652, −7.430546728953835, −6.613232957883877, −6.111921421661392, −5.282552713947827, −4.333408337525968, −3.864623848073164, −3.031546581412677, −2.311672140921618, −1.574276305940814, −0.5355267580567870,
0.5355267580567870, 1.574276305940814, 2.311672140921618, 3.031546581412677, 3.864623848073164, 4.333408337525968, 5.282552713947827, 6.111921421661392, 6.613232957883877, 7.430546728953835, 7.807905915524652, 8.422696942539207, 8.887940894967668, 9.527098494299203, 10.04837299742061, 10.66481580843974, 11.22166062198454, 11.82658911958211, 12.31697985296928, 13.00486023850522, 13.55918406892001, 14.19738450521627, 14.64197294125435, 15.44634616244571, 15.64685373784225
