| L(s) = 1 | + 2-s + 4-s − 5-s + 5·7-s + 8-s − 10-s + 5·13-s + 5·14-s + 16-s − 3·17-s − 7·19-s − 20-s + 23-s + 25-s + 5·26-s + 5·28-s + 6·29-s − 4·31-s + 32-s − 3·34-s − 5·35-s + 8·37-s − 7·38-s − 40-s + 12·41-s + 11·43-s + 46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.88·7-s + 0.353·8-s − 0.316·10-s + 1.38·13-s + 1.33·14-s + 1/4·16-s − 0.727·17-s − 1.60·19-s − 0.223·20-s + 0.208·23-s + 1/5·25-s + 0.980·26-s + 0.944·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.514·34-s − 0.845·35-s + 1.31·37-s − 1.13·38-s − 0.158·40-s + 1.87·41-s + 1.67·43-s + 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.081266587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.081266587\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 23 | \( 1 - T \) | |
| good | 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−7.85049839516216548268277781283, −7.57091909614049087123107923505, −6.35353134432663234639384097910, −6.02429223192644005723950145228, −4.94665357849859923774402461294, −4.35258373244961525418974463429, −4.02523299602352728998638999598, −2.74956946523248293562135130742, −1.91742496956265609693127419530, −1.01792245385563102027986754862,
1.01792245385563102027986754862, 1.91742496956265609693127419530, 2.74956946523248293562135130742, 4.02523299602352728998638999598, 4.35258373244961525418974463429, 4.94665357849859923774402461294, 6.02429223192644005723950145228, 6.35353134432663234639384097910, 7.57091909614049087123107923505, 7.85049839516216548268277781283
