| L(s) = 1 | + 3-s − 4·5-s + 9-s − 6·11-s − 13-s − 4·15-s − 6·17-s − 8·23-s + 11·25-s + 27-s + 6·29-s − 10·31-s − 6·33-s − 10·37-s − 39-s − 10·41-s + 8·43-s − 4·45-s + 12·47-s − 7·49-s − 6·51-s + 10·53-s + 24·55-s + 4·59-s + 6·61-s + 4·65-s + 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 1/3·9-s − 1.80·11-s − 0.277·13-s − 1.03·15-s − 1.45·17-s − 1.66·23-s + 11/5·25-s + 0.192·27-s + 1.11·29-s − 1.79·31-s − 1.04·33-s − 1.64·37-s − 0.160·39-s − 1.56·41-s + 1.21·43-s − 0.596·45-s + 1.75·47-s − 49-s − 0.840·51-s + 1.37·53-s + 3.23·55-s + 0.520·59-s + 0.768·61-s + 0.496·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112632 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112632 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−13.76126143482440, −13.49393294154858, −12.77534609423262, −12.41722356988969, −12.05622024383126, −11.45329060819440, −10.87546349288923, −10.54739701805338, −10.13949904495693, −9.356370532198902, −8.645345187056219, −8.438178091385863, −7.971514370256014, −7.518261283809356, −6.978772710811719, −6.706815997077598, −5.496114865053578, −5.306376327639130, −4.436783328839419, −4.108384072809962, −3.613011051613152, −2.902545532317396, −2.397498021375909, −1.801710162716378, −0.5056432001902866, 0,
0.5056432001902866, 1.801710162716378, 2.397498021375909, 2.902545532317396, 3.613011051613152, 4.108384072809962, 4.436783328839419, 5.306376327639130, 5.496114865053578, 6.706815997077598, 6.978772710811719, 7.518261283809356, 7.971514370256014, 8.438178091385863, 8.645345187056219, 9.356370532198902, 10.13949904495693, 10.54739701805338, 10.87546349288923, 11.45329060819440, 12.05622024383126, 12.41722356988969, 12.77534609423262, 13.49393294154858, 13.76126143482440
