| L(s) = 1 | + 3-s − 2·4-s − 2·9-s − 3·11-s − 2·12-s + 5·13-s + 4·16-s + 3·17-s − 2·19-s + 6·23-s − 5·27-s + 3·29-s + 4·31-s − 3·33-s + 4·36-s − 2·37-s + 5·39-s + 12·41-s + 10·43-s + 6·44-s + 9·47-s + 4·48-s + 3·51-s − 10·52-s − 12·53-s − 2·57-s − 8·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 2/3·9-s − 0.904·11-s − 0.577·12-s + 1.38·13-s + 16-s + 0.727·17-s − 0.458·19-s + 1.25·23-s − 0.962·27-s + 0.557·29-s + 0.718·31-s − 0.522·33-s + 2/3·36-s − 0.328·37-s + 0.800·39-s + 1.87·41-s + 1.52·43-s + 0.904·44-s + 1.31·47-s + 0.577·48-s + 0.420·51-s − 1.38·52-s − 1.64·53-s − 0.264·57-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.490882041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.490882041\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−9.408054549667512195696276237510, −8.942402513263511435033585555615, −8.173579333842050044011512625481, −7.65396811735679492201158087709, −6.18506570501734980145358588299, −5.49929554576320674834936734485, −4.50035894020182177306684850522, −3.50518882381413089746428044840, −2.67848601395212920300767663698, −0.913243850144851748702778420825,
0.913243850144851748702778420825, 2.67848601395212920300767663698, 3.50518882381413089746428044840, 4.50035894020182177306684850522, 5.49929554576320674834936734485, 6.18506570501734980145358588299, 7.65396811735679492201158087709, 8.173579333842050044011512625481, 8.942402513263511435033585555615, 9.408054549667512195696276237510
