| L(s) = 1 | − 5-s − 4·7-s − 3·9-s + 11-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 − 55-s + 4·59-s − 2·61-s + 12·63-s + 8·67-s + 14·73-s − 4·77-s − 16·79-s + 9·81-s + 8·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 9-s + 0.301·11-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 − 0.134·55-s + 0.520·59-s − 0.256·61-s + 1.51·63-s + 0.977·67-s + 1.63·73-s − 0.455·77-s − 1.80·79-s + 81-s + 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 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.k |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 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.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
−14.34770829279910, −13.78853863808435, −13.11411551755296, −12.85381253025164, −12.59656910691074, −11.61733475622406, −11.39471335276297, −10.91948294869487, −10.35745405130108, −9.670860456185036, −9.140553475110615, −8.911910936511003, −8.296709897011443, −7.678284554300514, −6.962187073682659, −6.416948659432329, −6.343674677994906, −5.464114235229357, −4.883125315666872, −4.150182891374540, −3.599134112739104, −3.104875603168954, −2.485872204399134, −1.813368690510553, −0.5635431551902857, 0,
0.5635431551902857, 1.813368690510553, 2.485872204399134, 3.104875603168954, 3.599134112739104, 4.150182891374540, 4.883125315666872, 5.464114235229357, 6.343674677994906, 6.416948659432329, 6.962187073682659, 7.678284554300514, 8.296709897011443, 8.911910936511003, 9.140553475110615, 9.670860456185036, 10.35745405130108, 10.91948294869487, 11.39471335276297, 11.61733475622406, 12.59656910691074, 12.85381253025164, 13.11411551755296, 13.78853863808435, 14.34770829279910
