| L(s) = 1 | + 3-s + 3·5-s + 7-s + 9-s − 3·11-s + 2·13-s + 3·15-s − 2·19-s + 21-s − 4·23-s + 4·25-s + 27-s + 3·29-s − 3·31-s − 3·33-s + 3·35-s + 6·37-s + 2·39-s − 4·41-s + 10·43-s + 3·45-s + 8·47-s + 49-s − 5·53-s − 9·55-s − 2·57-s − 13·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s + 0.554·13-s + 0.774·15-s − 0.458·19-s + 0.218·21-s − 0.834·23-s + 4/5·25-s + 0.192·27-s + 0.557·29-s − 0.538·31-s − 0.522·33-s + 0.507·35-s + 0.986·37-s + 0.320·39-s − 0.624·41-s + 1.52·43-s + 0.447·45-s + 1.16·47-s + 1/7·49-s − 0.686·53-s − 1.21·55-s − 0.264·57-s − 1.69·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.227329763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.227329763\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 + 13 T + p T^{2} \) | 1.59.n |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−14.33483715397835, −14.00351102953896, −13.68727717392647, −13.03722664022692, −12.74143994230541, −12.13675074965872, −11.32780994645296, −10.81416617607120, −10.25459723063670, −10.02397398376009, −9.141784747429302, −9.005617208546999, −8.243202663900428, −7.728170133195660, −7.268668682663444, −6.304329463550204, −6.022157794122546, −5.493035195458148, −4.708126454252473, −4.256060458939224, −3.353148628379159, −2.699624118897452, −2.103296801862404, −1.648341227784230, −0.6826361207113849,
0.6826361207113849, 1.648341227784230, 2.103296801862404, 2.699624118897452, 3.353148628379159, 4.256060458939224, 4.708126454252473, 5.493035195458148, 6.022157794122546, 6.304329463550204, 7.268668682663444, 7.728170133195660, 8.243202663900428, 9.005617208546999, 9.141784747429302, 10.02397398376009, 10.25459723063670, 10.81416617607120, 11.32780994645296, 12.13675074965872, 12.74143994230541, 13.03722664022692, 13.68727717392647, 14.00351102953896, 14.33483715397835
