| L(s) = 1 | − 2·3-s + 2·7-s + 9-s + 4·11-s + 2·13-s − 4·17-s − 4·19-s − 4·21-s + 4·23-s + 4·27-s − 2·29-s − 8·33-s − 4·39-s + 6·41-s + 6·43-s + 6·47-s − 3·49-s + 8·51-s − 4·53-s + 8·57-s + 12·59-s + 14·61-s + 2·63-s + 8·67-s − 8·69-s − 14·73-s + 8·77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.970·17-s − 0.917·19-s − 0.872·21-s + 0.834·23-s + 0.769·27-s − 0.371·29-s − 1.39·33-s − 0.640·39-s + 0.937·41-s + 0.914·43-s + 0.875·47-s − 3/7·49-s + 1.12·51-s − 0.549·53-s + 1.05·57-s + 1.56·59-s + 1.79·61-s + 0.251·63-s + 0.977·67-s − 0.963·69-s − 1.63·73-s + 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.139970294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.139970294\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−13.25225519201967, −13.06780708122373, −12.47665879720266, −11.93903972829015, −11.39682710758696, −11.21516127196803, −10.87754095416116, −10.30133961368002, −9.656851496713853, −8.992924410990951, −8.678104340353278, −8.270796135546651, −7.392865898108025, −6.936910850572305, −6.518923184193347, −5.929172510807966, −5.636209251096352, −4.867001669342228, −4.427901545934699, −4.024673938762805, −3.296821073199956, −2.372474229988336, −1.852472234443023, −1.011571680292330, −0.5793237752874486,
0.5793237752874486, 1.011571680292330, 1.852472234443023, 2.372474229988336, 3.296821073199956, 4.024673938762805, 4.427901545934699, 4.867001669342228, 5.636209251096352, 5.929172510807966, 6.518923184193347, 6.936910850572305, 7.392865898108025, 8.270796135546651, 8.678104340353278, 8.992924410990951, 9.656851496713853, 10.30133961368002, 10.87754095416116, 11.21516127196803, 11.39682710758696, 11.93903972829015, 12.47665879720266, 13.06780708122373, 13.25225519201967
