| L(s) = 1 | − 3-s + 9-s − 4·13-s + 8·19-s + 4·23-s − 27-s − 2·29-s + 4·31-s + 8·37-s + 4·39-s + 2·41-s − 4·47-s − 7·49-s + 12·53-s − 8·57-s − 12·59-s − 2·61-s + 12·67-s − 4·69-s + 12·73-s − 12·79-s + 81-s + 8·83-s + 2·87-s + 6·89-s − 4·93-s + 101-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.10·13-s + 1.83·19-s + 0.834·23-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 1.31·37-s + 0.640·39-s + 0.312·41-s − 0.583·47-s − 49-s + 1.64·53-s − 1.05·57-s − 1.56·59-s − 0.256·61-s + 1.46·67-s − 0.481·69-s + 1.40·73-s − 1.35·79-s + 1/9·81-s + 0.878·83-s + 0.214·87-s + 0.635·89-s − 0.414·93-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.861520441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.861520441\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−14.97129685884239, −14.37413684851092, −13.87678666248229, −13.27285424210355, −12.77118976526751, −12.21702590507688, −11.71661646452827, −11.31945159019748, −10.76250939134437, −10.00924892750134, −9.635687120880673, −9.258391827312215, −8.401888435049177, −7.650263275025735, −7.415900980954077, −6.717407703037647, −6.126023347327047, −5.408431334802852, −4.979645611725095, −4.473573615243753, −3.566864337826248, −2.936859396325811, −2.244519399300978, −1.239443493403258, −0.5812589388576235,
0.5812589388576235, 1.239443493403258, 2.244519399300978, 2.936859396325811, 3.566864337826248, 4.473573615243753, 4.979645611725095, 5.408431334802852, 6.126023347327047, 6.717407703037647, 7.415900980954077, 7.650263275025735, 8.401888435049177, 9.258391827312215, 9.635687120880673, 10.00924892750134, 10.76250939134437, 11.31945159019748, 11.71661646452827, 12.21702590507688, 12.77118976526751, 13.27285424210355, 13.87678666248229, 14.37413684851092, 14.97129685884239
