| L(s) = 1 | − 3-s − 2·5-s − 2·7-s + 9-s − 4·11-s + 6·13-s + 2·15-s + 6·17-s + 19-s + 2·21-s + 6·23-s − 25-s − 27-s + 2·29-s + 4·33-s + 4·35-s − 8·37-s − 6·39-s − 2·41-s − 43-s − 2·45-s − 2·47-s − 3·49-s − 6·51-s + 8·53-s + 8·55-s − 57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 0.516·15-s + 1.45·17-s + 0.229·19-s + 0.436·21-s + 1.25·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.696·33-s + 0.676·35-s − 1.31·37-s − 0.960·39-s − 0.312·41-s − 0.152·43-s − 0.298·45-s − 0.291·47-s − 3/7·49-s − 0.840·51-s + 1.09·53-s + 1.07·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| 43 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.19794661640619, −14.74035098001171, −13.81218775586359, −13.43166207081725, −13.00623891593175, −12.35527164514329, −12.02542871946685, −11.33431411408591, −10.96334407312144, −10.30222771523963, −10.05311986557771, −9.226621036400214, −8.503836864855903, −8.204495354154746, −7.387106024752555, −7.121296515544263, −6.293091905634208, −5.766243933179030, −5.269473410950229, −4.614421975267235, −3.729507050723326, −3.364341879454761, −2.828680189529192, −1.572790274899245, −0.8445729317642846, 0,
0.8445729317642846, 1.572790274899245, 2.828680189529192, 3.364341879454761, 3.729507050723326, 4.614421975267235, 5.269473410950229, 5.766243933179030, 6.293091905634208, 7.121296515544263, 7.387106024752555, 8.204495354154746, 8.503836864855903, 9.226621036400214, 10.05311986557771, 10.30222771523963, 10.96334407312144, 11.33431411408591, 12.02542871946685, 12.35527164514329, 13.00623891593175, 13.43166207081725, 13.81218775586359, 14.74035098001171, 15.19794661640619
