| L(s) = 1 | − 2·4-s + 7-s − 4·11-s + 4·16-s − 4·17-s + 19-s + 4·23-s − 2·28-s − 4·29-s + 5·31-s − 2·37-s + 4·43-s + 8·44-s + 2·47-s + 49-s − 6·53-s − 12·59-s − 15·61-s − 8·64-s − 4·67-s + 8·68-s − 10·71-s − 3·73-s − 2·76-s − 4·77-s − 11·79-s + 12·83-s + ⋯ |
| L(s) = 1 | − 4-s + 0.377·7-s − 1.20·11-s + 16-s − 0.970·17-s + 0.229·19-s + 0.834·23-s − 0.377·28-s − 0.742·29-s + 0.898·31-s − 0.328·37-s + 0.609·43-s + 1.20·44-s + 0.291·47-s + 1/7·49-s − 0.824·53-s − 1.56·59-s − 1.92·61-s − 64-s − 0.488·67-s + 0.970·68-s − 1.18·71-s − 0.351·73-s − 0.229·76-s − 0.455·77-s − 1.23·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 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.ae |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 15 T + p T^{2} \) | 1.61.p |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−13.31101402845446, −12.84484448006517, −12.43659171072400, −11.99726772687562, −11.29660463374448, −10.89297302258913, −10.50500010594456, −10.09298967564469, −9.383692401127721, −9.113450543778519, −8.697241204218165, −8.095776665850370, −7.706465742131035, −7.363960950555791, −6.632508860211045, −6.031045341790430, −5.569064695253094, −5.024963506696571, −4.640041698692231, −4.265776264850771, −3.575148976858028, −2.844458939929504, −2.620907983231544, −1.597602252909672, −1.176161570804885, 0, 0,
1.176161570804885, 1.597602252909672, 2.620907983231544, 2.844458939929504, 3.575148976858028, 4.265776264850771, 4.640041698692231, 5.024963506696571, 5.569064695253094, 6.031045341790430, 6.632508860211045, 7.363960950555791, 7.706465742131035, 8.095776665850370, 8.697241204218165, 9.113450543778519, 9.383692401127721, 10.09298967564469, 10.50500010594456, 10.89297302258913, 11.29660463374448, 11.99726772687562, 12.43659171072400, 12.84484448006517, 13.31101402845446
