| L(s) = 1 | − 3-s − 4·5-s − 7-s + 9-s − 3·11-s − 5·13-s + 4·15-s + 7·17-s + 5·19-s + 21-s + 23-s + 11·25-s − 27-s − 8·29-s + 6·31-s + 3·33-s + 4·35-s + 37-s + 5·39-s − 2·41-s + 12·43-s − 4·45-s − 2·47-s − 6·49-s − 7·51-s − 53-s + 12·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s − 1.38·13-s + 1.03·15-s + 1.69·17-s + 1.14·19-s + 0.218·21-s + 0.208·23-s + 11/5·25-s − 0.192·27-s − 1.48·29-s + 1.07·31-s + 0.522·33-s + 0.676·35-s + 0.164·37-s + 0.800·39-s − 0.312·41-s + 1.82·43-s − 0.596·45-s − 0.291·47-s − 6/7·49-s − 0.980·51-s − 0.137·53-s + 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6337216869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6337216869\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−10.13453063025828401235802413110, −9.495399027905710074938485572113, −8.035285416978774481992436654500, −7.63523218338737891189277998358, −7.00056867701910294647186543544, −5.54401215981372403141830249075, −4.87762249458603618299483203096, −3.75737859938663287985963067129, −2.87455899464875715262215767066, −0.63511399953589095833800615301,
0.63511399953589095833800615301, 2.87455899464875715262215767066, 3.75737859938663287985963067129, 4.87762249458603618299483203096, 5.54401215981372403141830249075, 7.00056867701910294647186543544, 7.63523218338737891189277998358, 8.035285416978774481992436654500, 9.495399027905710074938485572113, 10.13453063025828401235802413110
