| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 2·7-s − 8-s + 9-s + 10-s − 12-s + 13-s − 2·14-s + 15-s + 16-s − 18-s + 6·19-s − 20-s − 2·21-s + 8·23-s + 24-s + 25-s − 26-s − 27-s + 2·28-s + 2·29-s − 30-s + 4·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.277·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s + 1.37·19-s − 0.223·20-s − 0.436·21-s + 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.377·28-s + 0.371·29-s − 0.182·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.029863390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.029863390\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−13.66879822961344, −12.88205542358424, −12.71853593975223, −11.85555605016094, −11.60977424591109, −11.22964504988771, −10.81788548281672, −10.23088839820022, −9.770026207130195, −9.085648186665710, −8.832895111439304, −8.079617519726122, −7.615532218381240, −7.392336989017638, −6.570067458075920, −6.277040260988744, −5.480173105210141, −4.907386346336733, −4.675203352595269, −3.727527794776835, −3.183344238634188, −2.552688892853879, −1.686238977545193, −0.9991576913862109, −0.6519004758029955,
0.6519004758029955, 0.9991576913862109, 1.686238977545193, 2.552688892853879, 3.183344238634188, 3.727527794776835, 4.675203352595269, 4.907386346336733, 5.480173105210141, 6.277040260988744, 6.570067458075920, 7.392336989017638, 7.615532218381240, 8.079617519726122, 8.832895111439304, 9.085648186665710, 9.770026207130195, 10.23088839820022, 10.81788548281672, 11.22964504988771, 11.60977424591109, 11.85555605016094, 12.71853593975223, 12.88205542358424, 13.66879822961344
