| L(s) = 1 | − 3-s + 2·7-s + 9-s − 4·11-s − 13-s − 4·19-s − 2·21-s − 2·23-s − 5·25-s − 27-s + 4·29-s − 2·31-s + 4·33-s + 39-s − 2·41-s − 4·43-s + 4·47-s − 3·49-s + 6·53-s + 4·57-s + 4·59-s − 4·61-s + 2·63-s − 4·67-s + 2·69-s − 2·71-s + 10·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.917·19-s − 0.436·21-s − 0.417·23-s − 25-s − 0.192·27-s + 0.742·29-s − 0.359·31-s + 0.696·33-s + 0.160·39-s − 0.312·41-s − 0.609·43-s + 0.583·47-s − 3/7·49-s + 0.824·53-s + 0.529·57-s + 0.520·59-s − 0.512·61-s + 0.251·63-s − 0.488·67-s + 0.240·69-s − 0.237·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6377231388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6377231388\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.00200201385415, −12.79651708791311, −12.01699302637923, −11.88735638149181, −11.22758606695588, −10.80349483178514, −10.38990638206494, −10.00169219530598, −9.482062116301028, −8.748424384432361, −8.262239449837357, −7.953053280630099, −7.378843553824326, −6.917085124522783, −6.251524010778093, −5.797247434405595, −5.266009489642633, −4.854646790442334, −4.328275375951974, −3.801622638175366, −3.020580990802522, −2.309227829939114, −1.938026860053978, −1.158754693481256, −0.2443752600103664,
0.2443752600103664, 1.158754693481256, 1.938026860053978, 2.309227829939114, 3.020580990802522, 3.801622638175366, 4.328275375951974, 4.854646790442334, 5.266009489642633, 5.797247434405595, 6.251524010778093, 6.917085124522783, 7.378843553824326, 7.953053280630099, 8.262239449837357, 8.748424384432361, 9.482062116301028, 10.00169219530598, 10.38990638206494, 10.80349483178514, 11.22758606695588, 11.88735638149181, 12.01699302637923, 12.79651708791311, 13.00200201385415
