| L(s) = 1 | − 3-s + 2·5-s + 4·7-s + 9-s + 2·13-s − 2·15-s − 17-s − 4·19-s − 4·21-s + 4·23-s − 25-s − 27-s − 2·29-s + 4·31-s + 8·35-s − 6·37-s − 2·39-s + 6·41-s − 4·43-s + 2·45-s + 9·49-s + 51-s + 6·53-s + 4·57-s − 4·59-s + 6·61-s + 4·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s − 0.242·17-s − 0.917·19-s − 0.872·21-s + 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 1.35·35-s − 0.986·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s + 0.298·45-s + 9/7·49-s + 0.140·51-s + 0.824·53-s + 0.529·57-s − 0.520·59-s + 0.768·61-s + 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 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 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.89350393107393, −13.54508581510753, −13.15236403011356, −12.50341337579209, −12.00264087057537, −11.47817986883151, −11.02923038251851, −10.67866242878522, −10.22721840742432, −9.599067221997334, −8.981952367059444, −8.586582011443137, −8.036490595441734, −7.522919348960974, −6.776969976123231, −6.440653350274080, −5.751289131322250, −5.310077636359482, −4.916243797792305, −4.223898197613272, −3.807551339245577, −2.717916483485097, −2.198104583165554, −1.495912008020413, −1.143964104044504, 0,
1.143964104044504, 1.495912008020413, 2.198104583165554, 2.717916483485097, 3.807551339245577, 4.223898197613272, 4.916243797792305, 5.310077636359482, 5.751289131322250, 6.440653350274080, 6.776969976123231, 7.522919348960974, 8.036490595441734, 8.586582011443137, 8.981952367059444, 9.599067221997334, 10.22721840742432, 10.67866242878522, 11.02923038251851, 11.47817986883151, 12.00264087057537, 12.50341337579209, 13.15236403011356, 13.54508581510753, 13.89350393107393
