| L(s) = 1 | − 3-s − 7-s − 2·9-s + 5·13-s + 2·19-s + 21-s − 6·23-s + 5·27-s + 4·31-s + 2·37-s − 5·39-s − 12·41-s − 8·43-s − 9·47-s − 6·49-s + 6·53-s − 2·57-s − 9·59-s − 8·61-s + 2·63-s + 4·67-s + 6·69-s + 9·71-s − 2·73-s − 79-s + 81-s − 18·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s + 1.38·13-s + 0.458·19-s + 0.218·21-s − 1.25·23-s + 0.962·27-s + 0.718·31-s + 0.328·37-s − 0.800·39-s − 1.87·41-s − 1.21·43-s − 1.31·47-s − 6/7·49-s + 0.824·53-s − 0.264·57-s − 1.17·59-s − 1.02·61-s + 0.251·63-s + 0.488·67-s + 0.722·69-s + 1.06·71-s − 0.234·73-s − 0.112·79-s + 1/9·81-s − 1.97·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5780806370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5780806370\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 83 | \( 1 + 18 T + p T^{2} \) | 1.83.s |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−13.58895294022961, −13.10398367332754, −12.45176819503875, −11.95970581378034, −11.58457327701277, −11.22123402483428, −10.65773714874279, −10.10347703850034, −9.788547296791519, −9.095183453273826, −8.501460328502649, −8.211090235408341, −7.727240677503166, −6.801189648414856, −6.386401891671353, −6.209420249453103, −5.405832378543528, −5.132967987515750, −4.351065538626568, −3.722871269633991, −3.216672156358290, −2.716697665773428, −1.716200176075539, −1.268414816150621, −0.2452988528627272,
0.2452988528627272, 1.268414816150621, 1.716200176075539, 2.716697665773428, 3.216672156358290, 3.722871269633991, 4.351065538626568, 5.132967987515750, 5.405832378543528, 6.209420249453103, 6.386401891671353, 6.801189648414856, 7.727240677503166, 8.211090235408341, 8.501460328502649, 9.095183453273826, 9.788547296791519, 10.10347703850034, 10.65773714874279, 11.22123402483428, 11.58457327701277, 11.95970581378034, 12.45176819503875, 13.10398367332754, 13.58895294022961
