| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 3·11-s − 5·13-s + 14-s + 16-s − 17-s − 6·19-s − 20-s − 3·22-s + 8·23-s − 4·25-s − 5·26-s + 28-s − 4·29-s − 6·31-s + 32-s − 34-s − 35-s − 5·37-s − 6·38-s − 40-s + 4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.904·11-s − 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 1.37·19-s − 0.223·20-s − 0.639·22-s + 1.66·23-s − 4/5·25-s − 0.980·26-s + 0.188·28-s − 0.742·29-s − 1.07·31-s + 0.176·32-s − 0.171·34-s − 0.169·35-s − 0.821·37-s − 0.973·38-s − 0.158·40-s + 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 - 9 T + p T^{2} \) | 1.79.aj |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−8.638994685144250058755560119869, −7.57301228525302142745708435155, −7.34505531857276165262607879293, −6.26154065509721020055388332299, −5.25040301016024486816261342026, −4.76079717195031085903989016093, −3.84664484475956969536598354260, −2.77187493497640080174593109372, −1.93783074616492210425537816742, 0,
1.93783074616492210425537816742, 2.77187493497640080174593109372, 3.84664484475956969536598354260, 4.76079717195031085903989016093, 5.25040301016024486816261342026, 6.26154065509721020055388332299, 7.34505531857276165262607879293, 7.57301228525302142745708435155, 8.638994685144250058755560119869
