| L(s) = 1 | + 3-s + 7-s + 9-s + 2·11-s − 13-s + 4·17-s − 4·19-s + 21-s − 2·23-s − 5·25-s + 27-s + 6·29-s + 2·33-s + 10·37-s − 39-s + 4·41-s + 4·43-s + 49-s + 4·51-s + 10·53-s − 4·57-s − 4·59-s − 2·61-s + 63-s − 2·69-s + 6·71-s + 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.277·13-s + 0.970·17-s − 0.917·19-s + 0.218·21-s − 0.417·23-s − 25-s + 0.192·27-s + 1.11·29-s + 0.348·33-s + 1.64·37-s − 0.160·39-s + 0.624·41-s + 0.609·43-s + 1/7·49-s + 0.560·51-s + 1.37·53-s − 0.529·57-s − 0.520·59-s − 0.256·61-s + 0.125·63-s − 0.240·69-s + 0.712·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.687163204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.687163204\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−8.098225820552997776823221528461, −7.958731194267174523471317397116, −6.96586725704390539804768312340, −6.20990946014147515150327787312, −5.45286722771906805968576134016, −4.38001540961947469936916075666, −3.93242944981291926283668604940, −2.83356599595692265996403299665, −2.03533904119451063971207377291, −0.929175110490288964280986467431,
0.929175110490288964280986467431, 2.03533904119451063971207377291, 2.83356599595692265996403299665, 3.93242944981291926283668604940, 4.38001540961947469936916075666, 5.45286722771906805968576134016, 6.20990946014147515150327787312, 6.96586725704390539804768312340, 7.958731194267174523471317397116, 8.098225820552997776823221528461
