| L(s) = 1 | + 3-s + 5-s + 9-s − 2·13-s + 15-s + 6·17-s + 4·19-s + 4·23-s + 25-s + 27-s + 2·29-s − 31-s − 2·37-s − 2·39-s − 6·41-s − 4·43-s + 45-s − 7·49-s + 6·51-s − 10·53-s + 4·57-s + 12·59-s + 10·61-s − 2·65-s + 4·67-s + 4·69-s + 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.554·13-s + 0.258·15-s + 1.45·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.179·31-s − 0.328·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s − 49-s + 0.840·51-s − 1.37·53-s + 0.529·57-s + 1.56·59-s + 1.28·61-s − 0.248·65-s + 0.488·67-s + 0.481·69-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.787709847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.787709847\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 31 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 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
−8.461064390626336338843179070222, −7.83626702755014814724924418425, −7.11876394007612079632680409709, −6.40700748858407037788751630983, −5.29521655002048635010793424231, −4.96136141853777835130856978968, −3.60840445640615665004095154229, −3.07886251818621786736478137164, −2.03675816674289281737558783038, −0.994808318405501324571666441447,
0.994808318405501324571666441447, 2.03675816674289281737558783038, 3.07886251818621786736478137164, 3.60840445640615665004095154229, 4.96136141853777835130856978968, 5.29521655002048635010793424231, 6.40700748858407037788751630983, 7.11876394007612079632680409709, 7.83626702755014814724924418425, 8.461064390626336338843179070222
