| L(s) = 1 | − 3-s + 5-s − 3·7-s + 9-s − 6·11-s − 15-s + 6·17-s + 3·21-s + 2·23-s + 25-s − 27-s + 8·29-s − 3·31-s + 6·33-s − 3·35-s − 6·37-s − 2·41-s + 11·43-s + 45-s − 12·47-s + 2·49-s − 6·51-s − 12·53-s − 6·55-s + 15·61-s − 3·63-s + 5·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s − 1.80·11-s − 0.258·15-s + 1.45·17-s + 0.654·21-s + 0.417·23-s + 1/5·25-s − 0.192·27-s + 1.48·29-s − 0.538·31-s + 1.04·33-s − 0.507·35-s − 0.986·37-s − 0.312·41-s + 1.67·43-s + 0.149·45-s − 1.75·47-s + 2/7·49-s − 0.840·51-s − 1.64·53-s − 0.809·55-s + 1.92·61-s − 0.377·63-s + 0.610·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 15 T + p T^{2} \) | 1.61.ap |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 + 15 T + p T^{2} \) | 1.79.p |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−16.82288430921750, −16.28744608783280, −15.73696257713993, −15.61271775360708, −14.43162806013866, −14.12026757782552, −13.20653910404219, −12.79091351935688, −12.55849111342896, −11.72111695443127, −10.94462175975862, −10.34658584318552, −9.981121012001313, −9.505399203619687, −8.546277484364320, −7.888615915744550, −7.255115862883736, −6.502106577654970, −5.972958525253095, −5.184962891157853, −4.930218923709811, −3.594403126479513, −3.044136682914772, −2.266597817597400, −1.019844258023239, 0,
1.019844258023239, 2.266597817597400, 3.044136682914772, 3.594403126479513, 4.930218923709811, 5.184962891157853, 5.972958525253095, 6.502106577654970, 7.255115862883736, 7.888615915744550, 8.546277484364320, 9.505399203619687, 9.981121012001313, 10.34658584318552, 10.94462175975862, 11.72111695443127, 12.55849111342896, 12.79091351935688, 13.20653910404219, 14.12026757782552, 14.43162806013866, 15.61271775360708, 15.73696257713993, 16.28744608783280, 16.82288430921750
