| L(s) = 1 | + 5-s + 2·7-s − 3·9-s − 11-s − 8·19-s + 8·23-s + 25-s − 10·29-s − 8·31-s + 2·35-s + 10·37-s − 2·41-s − 6·43-s − 3·45-s + 8·47-s − 3·49-s − 14·53-s − 55-s − 4·59-s − 10·61-s − 6·63-s + 4·67-s − 8·73-s − 2·77-s + 4·79-s + 9·81-s + 10·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 9-s − 0.301·11-s − 1.83·19-s + 1.66·23-s + 1/5·25-s − 1.85·29-s − 1.43·31-s + 0.338·35-s + 1.64·37-s − 0.312·41-s − 0.914·43-s − 0.447·45-s + 1.16·47-s − 3/7·49-s − 1.92·53-s − 0.134·55-s − 0.520·59-s − 1.28·61-s − 0.755·63-s + 0.488·67-s − 0.936·73-s − 0.227·77-s + 0.450·79-s + 81-s + 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−8.184249725851853803834532515265, −7.60671209192694772214603774398, −6.63150790425675055814778311059, −5.88922529047574957731066072353, −5.21298710087630634622403275624, −4.48156605392211559871508964750, −3.39055982896274013705720197906, −2.43781193753337672507746790032, −1.60990262564069096103734615354, 0,
1.60990262564069096103734615354, 2.43781193753337672507746790032, 3.39055982896274013705720197906, 4.48156605392211559871508964750, 5.21298710087630634622403275624, 5.88922529047574957731066072353, 6.63150790425675055814778311059, 7.60671209192694772214603774398, 8.184249725851853803834532515265
