| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s + 11-s + 13-s − 15-s − 17-s − 7·19-s + 21-s − 3·23-s − 4·25-s + 27-s − 3·29-s − 8·31-s + 33-s − 35-s + 7·37-s + 39-s + 8·41-s − 7·43-s − 45-s − 8·47-s + 49-s − 51-s − 10·53-s − 55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s − 0.258·15-s − 0.242·17-s − 1.60·19-s + 0.218·21-s − 0.625·23-s − 4/5·25-s + 0.192·27-s − 0.557·29-s − 1.43·31-s + 0.174·33-s − 0.169·35-s + 1.15·37-s + 0.160·39-s + 1.24·41-s − 1.06·43-s − 0.149·45-s − 1.16·47-s + 1/7·49-s − 0.140·51-s − 1.37·53-s − 0.134·55-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−7.973913739293717184551560753427, −7.53009057739503362129078370482, −6.52588954577257861619038664455, −5.94629182737895382004814273022, −4.81970973793571202639356177016, −4.09290328256093731010904961199, −3.53824776432238292422073951005, −2.34083657626928236435282597946, −1.61071430826990798152270709526, 0,
1.61071430826990798152270709526, 2.34083657626928236435282597946, 3.53824776432238292422073951005, 4.09290328256093731010904961199, 4.81970973793571202639356177016, 5.94629182737895382004814273022, 6.52588954577257861619038664455, 7.53009057739503362129078370482, 7.973913739293717184551560753427
