| L(s) = 1 | + 2-s − 4-s − 4·7-s − 3·8-s − 4·11-s − 2·13-s − 4·14-s − 16-s + 2·17-s − 19-s − 4·22-s − 4·23-s − 2·26-s + 4·28-s + 2·29-s + 5·32-s + 2·34-s + 6·37-s − 38-s + 6·41-s − 8·43-s + 4·44-s − 4·46-s − 12·47-s + 9·49-s + 2·52-s − 14·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.51·7-s − 1.06·8-s − 1.20·11-s − 0.554·13-s − 1.06·14-s − 1/4·16-s + 0.485·17-s − 0.229·19-s − 0.852·22-s − 0.834·23-s − 0.392·26-s + 0.755·28-s + 0.371·29-s + 0.883·32-s + 0.342·34-s + 0.986·37-s − 0.162·38-s + 0.937·41-s − 1.21·43-s + 0.603·44-s − 0.589·46-s − 1.75·47-s + 9/7·49-s + 0.277·52-s − 1.92·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8429111105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8429111105\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−8.232275819067947015722607300017, −7.78209523078525975761964928898, −6.58687864019063039883776727456, −6.19991214263567143151204518968, −5.29946016368754541744477094849, −4.74301849189419911948357547981, −3.71181161492563870329011762014, −3.13512698083875842512786506180, −2.35078429164942207821577862739, −0.43645053784846297163211134797,
0.43645053784846297163211134797, 2.35078429164942207821577862739, 3.13512698083875842512786506180, 3.71181161492563870329011762014, 4.74301849189419911948357547981, 5.29946016368754541744477094849, 6.19991214263567143151204518968, 6.58687864019063039883776727456, 7.78209523078525975761964928898, 8.232275819067947015722607300017
