| L(s) = 1 | − 5-s + 6·11-s + 2·13-s + 17-s − 8·19-s + 4·23-s + 25-s − 6·29-s + 10·31-s − 6·37-s + 10·41-s − 2·43-s + 6·47-s − 7·49-s + 10·53-s − 6·55-s + 2·61-s − 2·65-s − 10·67-s − 14·71-s − 6·73-s + 10·79-s + 14·83-s − 85-s + 10·89-s + 8·95-s + 14·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.80·11-s + 0.554·13-s + 0.242·17-s − 1.83·19-s + 0.834·23-s + 1/5·25-s − 1.11·29-s + 1.79·31-s − 0.986·37-s + 1.56·41-s − 0.304·43-s + 0.875·47-s − 49-s + 1.37·53-s − 0.809·55-s + 0.256·61-s − 0.248·65-s − 1.22·67-s − 1.66·71-s − 0.702·73-s + 1.12·79-s + 1.53·83-s − 0.108·85-s + 1.05·89-s + 0.820·95-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.894749373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.894749373\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−8.900235850631111027856282739430, −8.040168400327605493833745110075, −7.10988674697168939663002572067, −6.46263712186345460804680945638, −5.87246175426078544956440123866, −4.57218762580994756696168648294, −4.06532761343116357519759891561, −3.24152080236141607595812306677, −1.96480032138734346021133671302, −0.870521071927731545118184019200,
0.870521071927731545118184019200, 1.96480032138734346021133671302, 3.24152080236141607595812306677, 4.06532761343116357519759891561, 4.57218762580994756696168648294, 5.87246175426078544956440123866, 6.46263712186345460804680945638, 7.10988674697168939663002572067, 8.040168400327605493833745110075, 8.900235850631111027856282739430
