| L(s) = 1 | − 3-s + 2·7-s + 9-s + 3·11-s + 13-s − 17-s − 2·21-s − 3·23-s − 5·25-s − 27-s + 8·29-s − 9·31-s − 3·33-s − 2·37-s − 39-s − 9·41-s + 43-s − 3·49-s + 51-s − 5·53-s + 4·59-s − 10·61-s + 2·63-s − 5·67-s + 3·69-s − 6·71-s + 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s − 0.242·17-s − 0.436·21-s − 0.625·23-s − 25-s − 0.192·27-s + 1.48·29-s − 1.61·31-s − 0.522·33-s − 0.328·37-s − 0.160·39-s − 1.40·41-s + 0.152·43-s − 3/7·49-s + 0.140·51-s − 0.686·53-s + 0.520·59-s − 1.28·61-s + 0.251·63-s − 0.610·67-s + 0.361·69-s − 0.712·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 43 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 9 T + p T^{2} \) | 1.31.j |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 13 T + p T^{2} \) | 1.83.n |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−7.38954097148291892565464181105, −6.72733933004863569120397622285, −6.07587561498051181580213997079, −5.42393405326039415285183687637, −4.62341170014414364044356269409, −4.05158513466036545936684319737, −3.20614858287851831712025235063, −1.92326413179271717739823232499, −1.36205032878156422288174029675, 0,
1.36205032878156422288174029675, 1.92326413179271717739823232499, 3.20614858287851831712025235063, 4.05158513466036545936684319737, 4.62341170014414364044356269409, 5.42393405326039415285183687637, 6.07587561498051181580213997079, 6.72733933004863569120397622285, 7.38954097148291892565464181105
