| L(s) = 1 | − 3·3-s + 5-s + 3·7-s + 6·9-s − 6·11-s − 13-s − 3·15-s + 17-s − 9·21-s + 6·23-s − 4·25-s − 9·27-s + 10·29-s + 18·33-s + 3·35-s + 9·37-s + 3·39-s − 4·41-s − 9·43-s + 6·45-s − 9·47-s + 2·49-s − 3·51-s − 2·53-s − 6·55-s + 12·61-s + 18·63-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 1.13·7-s + 2·9-s − 1.80·11-s − 0.277·13-s − 0.774·15-s + 0.242·17-s − 1.96·21-s + 1.25·23-s − 4/5·25-s − 1.73·27-s + 1.85·29-s + 3.13·33-s + 0.507·35-s + 1.47·37-s + 0.480·39-s − 0.624·41-s − 1.37·43-s + 0.894·45-s − 1.31·47-s + 2/7·49-s − 0.420·51-s − 0.274·53-s − 0.809·55-s + 1.53·61-s + 2.26·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.023890165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.023890165\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−8.353275643029372910992464360581, −7.88108031297765732460961197524, −6.97293775212790592999776217673, −6.27171944313527241602021102969, −5.32996229839017358192429773862, −5.08805445642714262648430614926, −4.48898825125575960761780679897, −2.90400787771345389296376415342, −1.78105435732361087376191105670, −0.66938611441662014856800735836,
0.66938611441662014856800735836, 1.78105435732361087376191105670, 2.90400787771345389296376415342, 4.48898825125575960761780679897, 5.08805445642714262648430614926, 5.32996229839017358192429773862, 6.27171944313527241602021102969, 6.97293775212790592999776217673, 7.88108031297765732460961197524, 8.353275643029372910992464360581
