| L(s) = 1 | − 2-s + 4-s − 2·5-s + 7-s − 8-s + 2·10-s − 11-s + 4·13-s − 14-s + 16-s + 2·17-s + 19-s − 2·20-s + 22-s − 25-s − 4·26-s + 28-s − 4·31-s − 32-s − 2·34-s − 2·35-s − 2·37-s − 38-s + 2·40-s + 2·41-s − 9·43-s − 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s − 0.353·8-s + 0.632·10-s − 0.301·11-s + 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.229·19-s − 0.447·20-s + 0.213·22-s − 1/5·25-s − 0.784·26-s + 0.188·28-s − 0.718·31-s − 0.176·32-s − 0.342·34-s − 0.338·35-s − 0.328·37-s − 0.162·38-s + 0.316·40-s + 0.312·41-s − 1.37·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−13.75406927172139, −13.07772288857149, −12.69285511586301, −12.02710951429695, −11.69029997766172, −11.25656845885456, −10.81283716553538, −10.35950553941539, −9.865237163994741, −9.127466916774695, −8.859114018800781, −8.244810616806189, −7.755954616989975, −7.612791014232632, −6.879781431720947, −6.314365721075522, −5.791221292658238, −5.205927853710245, −4.565439034167096, −3.946081479942895, −3.353739845139212, −3.012585881798184, −1.975569908034514, −1.530059535636023, −0.7521910573453265, 0,
0.7521910573453265, 1.530059535636023, 1.975569908034514, 3.012585881798184, 3.353739845139212, 3.946081479942895, 4.565439034167096, 5.205927853710245, 5.791221292658238, 6.314365721075522, 6.879781431720947, 7.612791014232632, 7.755954616989975, 8.244810616806189, 8.859114018800781, 9.127466916774695, 9.865237163994741, 10.35950553941539, 10.81283716553538, 11.25656845885456, 11.69029997766172, 12.02710951429695, 12.69285511586301, 13.07772288857149, 13.75406927172139
