| L(s) = 1 | − 5-s + 4·7-s − 3·9-s − 4·13-s + 4·17-s + 8·19-s − 4·23-s + 25-s + 8·29-s + 4·31-s − 4·35-s + 6·37-s + 8·41-s + 4·43-s + 3·45-s − 12·47-s + 9·49-s − 10·53-s − 8·61-s − 12·63-s + 4·65-s − 8·67-s − 12·71-s − 12·73-s − 8·79-s + 9·81-s + 4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 9-s − 1.10·13-s + 0.970·17-s + 1.83·19-s − 0.834·23-s + 1/5·25-s + 1.48·29-s + 0.718·31-s − 0.676·35-s + 0.986·37-s + 1.24·41-s + 0.609·43-s + 0.447·45-s − 1.75·47-s + 9/7·49-s − 1.37·53-s − 1.02·61-s − 1.51·63-s + 0.496·65-s − 0.977·67-s − 1.42·71-s − 1.40·73-s − 0.900·79-s + 81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.152572046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.152572046\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−7.76250582206361158704588981631, −7.35245225426805282507395824352, −6.13562083178715160063782914649, −5.61632446806025604274636075432, −4.67528474825564873970894075477, −4.61074666782449575769483425272, −3.19211400016719002818029975994, −2.78495535899106050514771359081, −1.63880190761205171043873480529, −0.72614825597823459730376568587,
0.72614825597823459730376568587, 1.63880190761205171043873480529, 2.78495535899106050514771359081, 3.19211400016719002818029975994, 4.61074666782449575769483425272, 4.67528474825564873970894075477, 5.61632446806025604274636075432, 6.13562083178715160063782914649, 7.35245225426805282507395824352, 7.76250582206361158704588981631
