| L(s) = 1 | − 3-s − 4·7-s + 9-s − 2·11-s − 13-s − 6·17-s − 4·19-s + 4·21-s + 4·23-s − 5·25-s − 27-s + 10·29-s − 8·31-s + 2·33-s − 2·37-s + 39-s − 4·43-s + 2·47-s + 9·49-s + 6·51-s − 2·53-s + 4·57-s + 10·59-s + 10·61-s − 4·63-s + 8·67-s − 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 1.45·17-s − 0.917·19-s + 0.872·21-s + 0.834·23-s − 25-s − 0.192·27-s + 1.85·29-s − 1.43·31-s + 0.348·33-s − 0.328·37-s + 0.160·39-s − 0.609·43-s + 0.291·47-s + 9/7·49-s + 0.840·51-s − 0.274·53-s + 0.529·57-s + 1.30·59-s + 1.28·61-s − 0.503·63-s + 0.977·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−11.11290586484749740754767267156, −10.32415831794851411382718499403, −9.482000907962227706474121895756, −8.460024182370040287671287102115, −6.97842775016457161547383853129, −6.43898703960602353565153271672, −5.24673574244182801120877493365, −3.96331013715912131044174768333, −2.52521406720822043214156379075, 0,
2.52521406720822043214156379075, 3.96331013715912131044174768333, 5.24673574244182801120877493365, 6.43898703960602353565153271672, 6.97842775016457161547383853129, 8.460024182370040287671287102115, 9.482000907962227706474121895756, 10.32415831794851411382718499403, 11.11290586484749740754767267156
