| L(s) = 1 | − 3-s + 3·5-s + 7-s + 9-s + 13-s − 3·15-s + 5·19-s − 21-s − 6·23-s + 4·25-s − 27-s + 3·29-s + 4·31-s + 3·35-s − 7·37-s − 39-s + 12·41-s + 2·43-s + 3·45-s − 3·47-s + 49-s + 6·53-s − 5·57-s − 3·59-s − 2·61-s + 63-s + 3·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 0.377·7-s + 1/3·9-s + 0.277·13-s − 0.774·15-s + 1.14·19-s − 0.218·21-s − 1.25·23-s + 4/5·25-s − 0.192·27-s + 0.557·29-s + 0.718·31-s + 0.507·35-s − 1.15·37-s − 0.160·39-s + 1.87·41-s + 0.304·43-s + 0.447·45-s − 0.437·47-s + 1/7·49-s + 0.824·53-s − 0.662·57-s − 0.390·59-s − 0.256·61-s + 0.125·63-s + 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.038871927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.038871927\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−14.55365286703877, −14.03927606737527, −13.91359363162941, −13.23262611895185, −12.75847807456870, −11.97106840331010, −11.82824089554752, −11.02466391114304, −10.51205376073006, −10.03018625115939, −9.641018167805443, −9.024073741246939, −8.446364687886432, −7.698186273370512, −7.233674218963416, −6.420835047894760, −5.989429992469176, −5.608924958924197, −4.959113304067275, −4.393892497886684, −3.590191094400234, −2.751442487149590, −2.061858036169639, −1.424091113586270, −0.6874029519276988,
0.6874029519276988, 1.424091113586270, 2.061858036169639, 2.751442487149590, 3.590191094400234, 4.393892497886684, 4.959113304067275, 5.608924958924197, 5.989429992469176, 6.420835047894760, 7.233674218963416, 7.698186273370512, 8.446364687886432, 9.024073741246939, 9.641018167805443, 10.03018625115939, 10.51205376073006, 11.02466391114304, 11.82824089554752, 11.97106840331010, 12.75847807456870, 13.23262611895185, 13.91359363162941, 14.03927606737527, 14.55365286703877
