| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 2·11-s − 4·13-s + 15-s + 2·17-s + 2·19-s + 21-s − 4·23-s + 25-s − 27-s − 6·29-s + 2·31-s + 2·33-s + 35-s − 10·37-s + 4·39-s − 10·41-s + 12·43-s − 45-s + 8·47-s + 49-s − 2·51-s + 2·55-s − 2·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s + 0.258·15-s + 0.485·17-s + 0.458·19-s + 0.218·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.359·31-s + 0.348·33-s + 0.169·35-s − 1.64·37-s + 0.640·39-s − 1.56·41-s + 1.82·43-s − 0.149·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s + 0.269·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6923238545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6923238545\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−7.68644732226053446030767385232, −7.41133995006281390270560894447, −6.64350677810854885419092220353, −5.70522143840793037649067280854, −5.27264772068337911049515166811, −4.43621706165973651292976079208, −3.63357330715571608109874316024, −2.79001339268664925957019516982, −1.78965542557283848314017161132, −0.42588739009589384131863225836,
0.42588739009589384131863225836, 1.78965542557283848314017161132, 2.79001339268664925957019516982, 3.63357330715571608109874316024, 4.43621706165973651292976079208, 5.27264772068337911049515166811, 5.70522143840793037649067280854, 6.64350677810854885419092220353, 7.41133995006281390270560894447, 7.68644732226053446030767385232
