| L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s − 2·11-s + 2·13-s − 15-s − 4·17-s − 2·19-s − 2·21-s + 25-s + 27-s + 4·29-s − 2·33-s + 2·35-s + 6·37-s + 2·39-s + 41-s + 4·43-s − 45-s + 6·47-s − 3·49-s − 4·51-s − 10·53-s + 2·55-s − 2·57-s + 14·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 0.258·15-s − 0.970·17-s − 0.458·19-s − 0.436·21-s + 1/5·25-s + 0.192·27-s + 0.742·29-s − 0.348·33-s + 0.338·35-s + 0.986·37-s + 0.320·39-s + 0.156·41-s + 0.609·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s − 0.560·51-s − 1.37·53-s + 0.269·55-s − 0.264·57-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 41 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 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.o |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−15.17963455823802, −14.44887165888924, −14.13021996206835, −13.36072254823586, −12.93806036445964, −12.77409774241945, −11.96089040326621, −11.31309765054255, −10.91256338499061, −10.24611315507456, −9.819735443476719, −9.112048132763373, −8.688072830572169, −8.174440085061195, −7.602187216757869, −6.985925670323683, −6.425596873489823, −5.930557292046564, −5.090869524330120, −4.318845541270935, −4.003862372600250, −3.105458363865747, −2.712162019292445, −1.955664861333013, −0.9223349175415376, 0,
0.9223349175415376, 1.955664861333013, 2.712162019292445, 3.105458363865747, 4.003862372600250, 4.318845541270935, 5.090869524330120, 5.930557292046564, 6.425596873489823, 6.985925670323683, 7.602187216757869, 8.174440085061195, 8.688072830572169, 9.112048132763373, 9.819735443476719, 10.24611315507456, 10.91256338499061, 11.31309765054255, 11.96089040326621, 12.77409774241945, 12.93806036445964, 13.36072254823586, 14.13021996206835, 14.44887165888924, 15.17963455823802
