| L(s) = 1 | + 2·3-s + 5-s − 2·7-s + 9-s − 11-s − 6·13-s + 2·15-s − 2·17-s − 8·19-s − 4·21-s + 25-s − 4·27-s + 2·29-s − 2·33-s − 2·35-s − 10·37-s − 12·39-s + 8·41-s + 8·43-s + 45-s − 12·47-s − 3·49-s − 4·51-s + 6·53-s − 55-s − 16·57-s − 8·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 0.516·15-s − 0.485·17-s − 1.83·19-s − 0.872·21-s + 1/5·25-s − 0.769·27-s + 0.371·29-s − 0.348·33-s − 0.338·35-s − 1.64·37-s − 1.92·39-s + 1.24·41-s + 1.21·43-s + 0.149·45-s − 1.75·47-s − 3/7·49-s − 0.560·51-s + 0.824·53-s − 0.134·55-s − 2.11·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.375799726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.375799726\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 79 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−14.81486236141061, −14.52752086005140, −13.89105667302736, −13.51104191820619, −12.79938584070942, −12.56720523912687, −12.04904384211530, −11.03497440796824, −10.66985925383227, −9.918476354297460, −9.643142736263907, −8.990622801596417, −8.653962196867132, −7.923282680819101, −7.465406425172671, −6.733249528547607, −6.319521972014044, −5.562018057075762, −4.766712197951156, −4.303514401783848, −3.408910207550938, −2.883705101541185, −2.216142786910453, −1.924464636528842, −0.3680656267421439,
0.3680656267421439, 1.924464636528842, 2.216142786910453, 2.883705101541185, 3.408910207550938, 4.303514401783848, 4.766712197951156, 5.562018057075762, 6.319521972014044, 6.733249528547607, 7.465406425172671, 7.923282680819101, 8.653962196867132, 8.990622801596417, 9.643142736263907, 9.918476354297460, 10.66985925383227, 11.03497440796824, 12.04904384211530, 12.56720523912687, 12.79938584070942, 13.51104191820619, 13.89105667302736, 14.52752086005140, 14.81486236141061
