| L(s) = 1 | − 5-s − 7-s + 3·11-s − 2·13-s − 3·17-s − 2·19-s + 25-s − 3·29-s − 31-s + 35-s − 37-s − 9·41-s − 11·43-s − 6·49-s − 9·53-s − 3·55-s − 6·59-s + 61-s + 2·65-s − 8·67-s + 12·71-s + 8·73-s − 3·77-s − 4·79-s − 6·83-s + 3·85-s + 6·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.904·11-s − 0.554·13-s − 0.727·17-s − 0.458·19-s + 1/5·25-s − 0.557·29-s − 0.179·31-s + 0.169·35-s − 0.164·37-s − 1.40·41-s − 1.67·43-s − 6/7·49-s − 1.23·53-s − 0.404·55-s − 0.781·59-s + 0.128·61-s + 0.248·65-s − 0.977·67-s + 1.42·71-s + 0.936·73-s − 0.341·77-s − 0.450·79-s − 0.658·83-s + 0.325·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2847621721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2847621721\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 37 | \( 1 + T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−13.60755485004924, −13.22179269839967, −12.69800615575691, −12.12143236899660, −11.88342752198650, −11.17671618959883, −10.92276404755373, −10.22966626941400, −9.622827799250559, −9.376964157545288, −8.701469337032288, −8.251337329276619, −7.771707248819600, −6.983287798713199, −6.697626229514984, −6.312997535370644, −5.502133127243376, −4.895515484854287, −4.460455600345223, −3.748247108080251, −3.369784047546824, −2.664771793426295, −1.857670117327449, −1.368764750020465, −0.1622434275610257,
0.1622434275610257, 1.368764750020465, 1.857670117327449, 2.664771793426295, 3.369784047546824, 3.748247108080251, 4.460455600345223, 4.895515484854287, 5.502133127243376, 6.312997535370644, 6.697626229514984, 6.983287798713199, 7.771707248819600, 8.251337329276619, 8.701469337032288, 9.376964157545288, 9.622827799250559, 10.22966626941400, 10.92276404755373, 11.17671618959883, 11.88342752198650, 12.12143236899660, 12.69800615575691, 13.22179269839967, 13.60755485004924
