| L(s) = 1 | + 3-s + 9-s − 5·11-s − 4·13-s + 17-s − 19-s + 27-s + 9·29-s + 6·31-s − 5·33-s − 3·37-s − 4·39-s − 5·41-s + 2·43-s − 9·47-s + 51-s + 3·53-s − 57-s + 6·59-s − 14·67-s − 8·71-s − 7·73-s − 6·79-s + 81-s − 12·83-s + 9·87-s + 6·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.50·11-s − 1.10·13-s + 0.242·17-s − 0.229·19-s + 0.192·27-s + 1.67·29-s + 1.07·31-s − 0.870·33-s − 0.493·37-s − 0.640·39-s − 0.780·41-s + 0.304·43-s − 1.31·47-s + 0.140·51-s + 0.412·53-s − 0.132·57-s + 0.781·59-s − 1.71·67-s − 0.949·71-s − 0.819·73-s − 0.675·79-s + 1/9·81-s − 1.31·83-s + 0.964·87-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 249900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 249900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.545602709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.545602709\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−12.95293581513891, −12.44633595346278, −11.87756661460498, −11.69983471327774, −10.86114826303785, −10.28868520460475, −10.12053909911938, −9.840261366070117, −9.004515364099229, −8.587520634992312, −8.158176125684886, −7.794783279508274, −7.127591412019332, −6.966710003773749, −6.111497855472028, −5.718291708639769, −4.883850846799467, −4.782644464176146, −4.244129741088151, −3.222381939073034, −3.042421042158201, −2.464206544305958, −1.969591758368981, −1.167762420800534, −0.3290889375970002,
0.3290889375970002, 1.167762420800534, 1.969591758368981, 2.464206544305958, 3.042421042158201, 3.222381939073034, 4.244129741088151, 4.782644464176146, 4.883850846799467, 5.718291708639769, 6.111497855472028, 6.966710003773749, 7.127591412019332, 7.794783279508274, 8.158176125684886, 8.587520634992312, 9.004515364099229, 9.840261366070117, 10.12053909911938, 10.28868520460475, 10.86114826303785, 11.69983471327774, 11.87756661460498, 12.44633595346278, 12.95293581513891
