| L(s) = 1 | − 3-s − 2·7-s + 9-s − 6·17-s + 2·19-s + 2·21-s − 5·25-s − 27-s + 6·29-s − 2·31-s + 2·37-s + 12·41-s + 4·43-s − 3·49-s + 6·51-s − 6·53-s − 2·57-s + 12·59-s − 2·61-s − 2·63-s − 10·67-s − 12·71-s − 14·73-s + 5·75-s + 8·79-s + 81-s + 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.45·17-s + 0.458·19-s + 0.436·21-s − 25-s − 0.192·27-s + 1.11·29-s − 0.359·31-s + 0.328·37-s + 1.87·41-s + 0.609·43-s − 3/7·49-s + 0.840·51-s − 0.824·53-s − 0.264·57-s + 1.56·59-s − 0.256·61-s − 0.251·63-s − 1.22·67-s − 1.42·71-s − 1.63·73-s + 0.577·75-s + 0.900·79-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−15.53241286433757, −14.76760087670818, −14.27607667710180, −13.53402123600131, −13.18890249668717, −12.73620778396827, −12.04529886369554, −11.65813125246614, −11.02473922068182, −10.57550780401249, −9.980755572945561, −9.347923538124627, −9.046729460875612, −8.219980492982182, −7.574796827886111, −7.027594636933914, −6.378416332215352, −6.012066253732343, −5.386197206648730, −4.452950397001930, −4.247158378229496, −3.283653590232801, −2.624402518472954, −1.856836738129533, −0.8396564314423784, 0,
0.8396564314423784, 1.856836738129533, 2.624402518472954, 3.283653590232801, 4.247158378229496, 4.452950397001930, 5.386197206648730, 6.012066253732343, 6.378416332215352, 7.027594636933914, 7.574796827886111, 8.219980492982182, 9.046729460875612, 9.347923538124627, 9.980755572945561, 10.57550780401249, 11.02473922068182, 11.65813125246614, 12.04529886369554, 12.73620778396827, 13.18890249668717, 13.53402123600131, 14.27607667710180, 14.76760087670818, 15.53241286433757
