| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 11-s − 2·13-s + 15-s + 2·17-s − 21-s + 25-s − 27-s + 6·29-s + 8·31-s − 33-s − 35-s + 2·37-s + 2·39-s − 2·41-s + 4·43-s − 45-s + 49-s − 2·51-s + 10·53-s − 55-s + 4·59-s − 6·61-s + 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.218·21-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.174·33-s − 0.169·35-s + 0.328·37-s + 0.320·39-s − 0.312·41-s + 0.609·43-s − 0.149·45-s + 1/7·49-s − 0.280·51-s + 1.37·53-s − 0.134·55-s + 0.520·59-s − 0.768·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.106076183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.106076183\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 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.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 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.g |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.02353369747634, −13.67462226569542, −12.99617862193905, −12.43282391725933, −11.89868557836357, −11.79989159079352, −11.17910885914861, −10.46154044625479, −10.21218595662121, −9.672642918830819, −8.859716025699818, −8.578465258241294, −7.693504217993554, −7.586249865214125, −6.794973842939654, −6.312327536130167, −5.772117861333844, −5.053364041998657, −4.613361001061115, −4.161009872452286, −3.366726947501337, −2.736727280461010, −2.005010227953867, −1.086974255023124, −0.5854622036434491,
0.5854622036434491, 1.086974255023124, 2.005010227953867, 2.736727280461010, 3.366726947501337, 4.161009872452286, 4.613361001061115, 5.053364041998657, 5.772117861333844, 6.312327536130167, 6.794973842939654, 7.586249865214125, 7.693504217993554, 8.578465258241294, 8.859716025699818, 9.672642918830819, 10.21218595662121, 10.46154044625479, 11.17910885914861, 11.79989159079352, 11.89868557836357, 12.43282391725933, 12.99617862193905, 13.67462226569542, 14.02353369747634
