| L(s) = 1 | − 3-s + 5-s + 9-s − 4·11-s − 13-s − 15-s + 6·17-s − 8·19-s − 8·23-s + 25-s − 27-s + 2·29-s + 8·31-s + 4·33-s + 2·37-s + 39-s + 2·41-s + 8·43-s + 45-s − 12·47-s − 6·51-s + 10·53-s − 4·55-s + 8·57-s + 4·59-s − 2·61-s − 65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.258·15-s + 1.45·17-s − 1.83·19-s − 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.696·33-s + 0.328·37-s + 0.160·39-s + 0.312·41-s + 1.21·43-s + 0.149·45-s − 1.75·47-s − 0.840·51-s + 1.37·53-s − 0.539·55-s + 1.05·57-s + 0.520·59-s − 0.256·61-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.489830748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.489830748\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.21569794451543, −12.84007660512408, −12.22341403952958, −12.16320670313276, −11.35696413867424, −10.91623283852829, −10.24775959632300, −10.12345418803554, −9.794627521763302, −8.990150587584886, −8.281716046953303, −8.040939472496443, −7.592809942513739, −6.816077510151804, −6.356352136935447, −5.857809235741585, −5.525515344727351, −4.832436456113073, −4.425499708792574, −3.798342253832097, −3.061369326713613, −2.318320269075593, −2.078535974734194, −1.073306172410273, −0.4071527472569952,
0.4071527472569952, 1.073306172410273, 2.078535974734194, 2.318320269075593, 3.061369326713613, 3.798342253832097, 4.425499708792574, 4.832436456113073, 5.525515344727351, 5.857809235741585, 6.356352136935447, 6.816077510151804, 7.592809942513739, 8.040939472496443, 8.281716046953303, 8.990150587584886, 9.794627521763302, 10.12345418803554, 10.24775959632300, 10.91623283852829, 11.35696413867424, 12.16320670313276, 12.22341403952958, 12.84007660512408, 13.21569794451543
