| L(s) = 1 | + 3-s + 2·5-s + 2·7-s + 9-s − 11-s + 4·13-s + 2·15-s + 17-s + 6·19-s + 2·21-s + 6·23-s − 25-s + 27-s − 2·29-s + 4·31-s − 33-s + 4·35-s − 6·37-s + 4·39-s + 6·41-s − 2·43-s + 2·45-s − 12·47-s − 3·49-s + 51-s + 4·53-s − 2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 0.516·15-s + 0.242·17-s + 1.37·19-s + 0.436·21-s + 1.25·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.174·33-s + 0.676·35-s − 0.986·37-s + 0.640·39-s + 0.937·41-s − 0.304·43-s + 0.298·45-s − 1.75·47-s − 3/7·49-s + 0.140·51-s + 0.549·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.144271383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.144271383\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.80096283201595, −14.32154628818759, −13.86876213772235, −13.36265984515022, −13.13369425023002, −12.36268472460108, −11.63222553883048, −11.25516330637138, −10.64967306413210, −10.02771704515555, −9.620096369106369, −8.924203370706769, −8.627283596211022, −7.785277987373181, −7.567586538998651, −6.666342573760801, −6.170259340469910, −5.373680391056865, −5.077137364359678, −4.302879832397101, −3.361550340743669, −3.072952111238742, −2.079301214109354, −1.529202515657027, −0.8733757374180488,
0.8733757374180488, 1.529202515657027, 2.079301214109354, 3.072952111238742, 3.361550340743669, 4.302879832397101, 5.077137364359678, 5.373680391056865, 6.170259340469910, 6.666342573760801, 7.567586538998651, 7.785277987373181, 8.627283596211022, 8.924203370706769, 9.620096369106369, 10.02771704515555, 10.64967306413210, 11.25516330637138, 11.63222553883048, 12.36268472460108, 13.13369425023002, 13.36265984515022, 13.86876213772235, 14.32154628818759, 14.80096283201595
