| L(s) = 1 | − 2·3-s − 2·5-s + 9-s + 6·11-s − 4·13-s + 4·15-s + 2·17-s − 4·19-s − 23-s − 25-s + 4·27-s + 10·29-s − 8·31-s − 12·33-s + 8·37-s + 8·39-s + 2·41-s + 6·43-s − 2·45-s + 12·47-s − 4·51-s − 12·53-s − 12·55-s + 8·57-s + 6·59-s − 6·61-s + 8·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 1/3·9-s + 1.80·11-s − 1.10·13-s + 1.03·15-s + 0.485·17-s − 0.917·19-s − 0.208·23-s − 1/5·25-s + 0.769·27-s + 1.85·29-s − 1.43·31-s − 2.08·33-s + 1.31·37-s + 1.28·39-s + 0.312·41-s + 0.914·43-s − 0.298·45-s + 1.75·47-s − 0.560·51-s − 1.64·53-s − 1.61·55-s + 1.05·57-s + 0.781·59-s − 0.768·61-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−14.42650646827040, −14.13129170455640, −13.19394785615355, −12.48337247892786, −12.27235521458085, −11.83869002256350, −11.54152095921861, −10.90119515634648, −10.53508781930012, −9.877063355096174, −9.245921655859361, −8.882953596596022, −8.141253146774397, −7.584057152159892, −7.075909587068590, −6.545871381453049, −6.010561788231838, −5.637517027957068, −4.683594102141303, −4.388741021223834, −3.939800833738069, −3.110987335938034, −2.393439242922677, −1.421860144372031, −0.7509828077885423, 0,
0.7509828077885423, 1.421860144372031, 2.393439242922677, 3.110987335938034, 3.939800833738069, 4.388741021223834, 4.683594102141303, 5.637517027957068, 6.010561788231838, 6.545871381453049, 7.075909587068590, 7.584057152159892, 8.141253146774397, 8.882953596596022, 9.245921655859361, 9.877063355096174, 10.53508781930012, 10.90119515634648, 11.54152095921861, 11.83869002256350, 12.27235521458085, 12.48337247892786, 13.19394785615355, 14.13129170455640, 14.42650646827040
