| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s − 2·9-s + 10-s + 12-s − 13-s + 14-s + 15-s + 16-s + 3·17-s − 2·18-s + 7·19-s + 20-s + 21-s + 24-s + 25-s − 26-s − 5·27-s + 28-s + 3·29-s + 30-s + 5·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.471·18-s + 1.60·19-s + 0.223·20-s + 0.218·21-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.962·27-s + 0.188·28-s + 0.557·29-s + 0.182·30-s + 0.898·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.373249942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.373249942\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 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 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−15.71286411648899, −15.50158274572425, −14.59561584360353, −14.17607209965397, −13.96872356191081, −13.47920985584922, −12.60888517106774, −12.12628348639729, −11.67746981208431, −10.97330328779883, −10.40137688280344, −9.659674977836192, −9.222562427091947, −8.473794511688548, −7.786259334744195, −7.429937282408006, −6.543784674891132, −5.845953819248819, −5.330368086361768, −4.777336045549440, −3.868301926992755, −3.103274727390376, −2.706141506272403, −1.804780982273589, −0.8921346447480759,
0.8921346447480759, 1.804780982273589, 2.706141506272403, 3.103274727390376, 3.868301926992755, 4.777336045549440, 5.330368086361768, 5.845953819248819, 6.543784674891132, 7.429937282408006, 7.786259334744195, 8.473794511688548, 9.222562427091947, 9.659674977836192, 10.40137688280344, 10.97330328779883, 11.67746981208431, 12.12628348639729, 12.60888517106774, 13.47920985584922, 13.96872356191081, 14.17607209965397, 14.59561584360353, 15.50158274572425, 15.71286411648899
