| L(s) = 1 | − 5-s + 4·7-s − 3·9-s + 6·13-s + 6·17-s − 4·19-s − 4·23-s + 25-s − 2·29-s − 8·31-s − 4·35-s + 10·37-s − 10·41-s + 3·45-s − 4·47-s + 9·49-s + 10·53-s − 4·59-s − 2·61-s − 12·63-s − 6·65-s − 8·67-s + 14·73-s − 16·79-s + 9·81-s + 8·83-s − 6·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 9-s + 1.66·13-s + 1.45·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.676·35-s + 1.64·37-s − 1.56·41-s + 0.447·45-s − 0.583·47-s + 9/7·49-s + 1.37·53-s − 0.520·59-s − 0.256·61-s − 1.51·63-s − 0.744·65-s − 0.977·67-s + 1.63·73-s − 1.80·79-s + 81-s + 0.878·83-s − 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 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 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.93051498855903, −14.60580804351849, −14.20030186721502, −13.54073612261587, −13.15040082718953, −12.25712367920759, −11.90057594085590, −11.37068914618113, −10.92298033584111, −10.65880601083174, −9.807863518605175, −9.059966710476745, −8.504473697847993, −8.111935624140657, −7.868248351004810, −7.060593890993920, −6.235989113530083, −5.693824596575179, −5.335574287452042, −4.505830528326834, −3.854203963884768, −3.434387242830116, −2.513906626077019, −1.673678104618374, −1.140881549077235, 0,
1.140881549077235, 1.673678104618374, 2.513906626077019, 3.434387242830116, 3.854203963884768, 4.505830528326834, 5.335574287452042, 5.693824596575179, 6.235989113530083, 7.060593890993920, 7.868248351004810, 8.111935624140657, 8.504473697847993, 9.059966710476745, 9.807863518605175, 10.65880601083174, 10.92298033584111, 11.37068914618113, 11.90057594085590, 12.25712367920759, 13.15040082718953, 13.54073612261587, 14.20030186721502, 14.60580804351849, 14.93051498855903
